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Properties

Label 2-74-37.7-c7-0-0
Degree 22
Conductor 7474
Sign −0.926−0.377i-0.926 - 0.377i
Analytic cond. 23.116423.1164
Root an. cond. 4.807964.80796
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.51 − 2.73i)2-s + (−19.5 − 7.11i)3-s + (49.0 − 41.1i)4-s + (69.5 + 394. i)5-s − 166.·6-s + (−65.8 − 373. i)7-s + (256. − 443. i)8-s + (−1.34e3 − 1.12e3i)9-s + (1.60e3 + 2.77e3i)10-s + (−2.61e3 + 4.52e3i)11-s + (−1.25e3 + 455. i)12-s + (1.58e3 − 1.33e3i)13-s + (−1.51e3 − 2.62e3i)14-s + (1.44e3 − 8.20e3i)15-s + (711. − 4.03e3i)16-s + (−2.42e4 − 2.03e4i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (−0.417 − 0.152i)3-s + (0.383 − 0.321i)4-s + (0.248 + 1.41i)5-s − 0.314·6-s + (−0.0725 − 0.411i)7-s + (0.176 − 0.306i)8-s + (−0.614 − 0.515i)9-s + (0.506 + 0.877i)10-s + (−0.591 + 1.02i)11-s + (−0.208 + 0.0760i)12-s + (0.200 − 0.167i)13-s + (−0.147 − 0.255i)14-s + (0.110 − 0.627i)15-s + (0.0434 − 0.246i)16-s + (−1.19 − 1.00i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((−0.926−0.377i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.377i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=((−0.926−0.377i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: −0.926−0.377i-0.926 - 0.377i
Analytic conductor: 23.116423.1164
Root analytic conductor: 4.807964.80796
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ74(7,⋅)\chi_{74} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 74, ( :7/2), −0.926−0.377i)(2,\ 74,\ (\ :7/2),\ -0.926 - 0.377i)

Particular Values

L(4)L(4) ≈\approx 0.0748705+0.382387i0.0748705 + 0.382387i
L(12)L(\frac12) ≈\approx 0.0748705+0.382387i0.0748705 + 0.382387i
L(92)L(\frac{9}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−7.51+2.73i)T 1 + (-7.51 + 2.73i)T
37 1+(−2.87e5+1.11e5i)T 1 + (-2.87e5 + 1.11e5i)T
good3 1+(19.5+7.11i)T+(1.67e3+1.40e3i)T2 1 + (19.5 + 7.11i)T + (1.67e3 + 1.40e3i)T^{2}
5 1+(−69.5−394.i)T+(−7.34e4+2.67e4i)T2 1 + (-69.5 - 394. i)T + (-7.34e4 + 2.67e4i)T^{2}
7 1+(65.8+373.i)T+(−7.73e5+2.81e5i)T2 1 + (65.8 + 373. i)T + (-7.73e5 + 2.81e5i)T^{2}
11 1+(2.61e3−4.52e3i)T+(−9.74e6−1.68e7i)T2 1 + (2.61e3 - 4.52e3i)T + (-9.74e6 - 1.68e7i)T^{2}
13 1+(−1.58e3+1.33e3i)T+(1.08e7−6.17e7i)T2 1 + (-1.58e3 + 1.33e3i)T + (1.08e7 - 6.17e7i)T^{2}
17 1+(2.42e4+2.03e4i)T+(7.12e7+4.04e8i)T2 1 + (2.42e4 + 2.03e4i)T + (7.12e7 + 4.04e8i)T^{2}
19 1+(2.60e4+9.46e3i)T+(6.84e8+5.74e8i)T2 1 + (2.60e4 + 9.46e3i)T + (6.84e8 + 5.74e8i)T^{2}
23 1+(−1.55e4−2.68e4i)T+(−1.70e9+2.94e9i)T2 1 + (-1.55e4 - 2.68e4i)T + (-1.70e9 + 2.94e9i)T^{2}
29 1+(1.27e5−2.21e5i)T+(−8.62e9−1.49e10i)T2 1 + (1.27e5 - 2.21e5i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1+2.90e5T+2.75e10T2 1 + 2.90e5T + 2.75e10T^{2}
41 1+(3.76e5−3.15e5i)T+(3.38e10−1.91e11i)T2 1 + (3.76e5 - 3.15e5i)T + (3.38e10 - 1.91e11i)T^{2}
43 1+2.40e5T+2.71e11T2 1 + 2.40e5T + 2.71e11T^{2}
47 1+(1.96e5+3.40e5i)T+(−2.53e11+4.38e11i)T2 1 + (1.96e5 + 3.40e5i)T + (-2.53e11 + 4.38e11i)T^{2}
53 1+(1.48e5−8.42e5i)T+(−1.10e12−4.01e11i)T2 1 + (1.48e5 - 8.42e5i)T + (-1.10e12 - 4.01e11i)T^{2}
59 1+(−1.63e5+9.27e5i)T+(−2.33e12−8.51e11i)T2 1 + (-1.63e5 + 9.27e5i)T + (-2.33e12 - 8.51e11i)T^{2}
61 1+(−1.02e6+8.59e5i)T+(5.45e11−3.09e12i)T2 1 + (-1.02e6 + 8.59e5i)T + (5.45e11 - 3.09e12i)T^{2}
67 1+(6.68e5+3.79e6i)T+(−5.69e12+2.07e12i)T2 1 + (6.68e5 + 3.79e6i)T + (-5.69e12 + 2.07e12i)T^{2}
71 1+(2.35e6+8.55e5i)T+(6.96e12+5.84e12i)T2 1 + (2.35e6 + 8.55e5i)T + (6.96e12 + 5.84e12i)T^{2}
73 1−5.32e6T+1.10e13T2 1 - 5.32e6T + 1.10e13T^{2}
79 1+(1.92e5+1.09e6i)T+(−1.80e13+6.56e12i)T2 1 + (1.92e5 + 1.09e6i)T + (-1.80e13 + 6.56e12i)T^{2}
83 1+(−7.07e6−5.93e6i)T+(4.71e12+2.67e13i)T2 1 + (-7.07e6 - 5.93e6i)T + (4.71e12 + 2.67e13i)T^{2}
89 1+(2.16e6−1.22e7i)T+(−4.15e13−1.51e13i)T2 1 + (2.16e6 - 1.22e7i)T + (-4.15e13 - 1.51e13i)T^{2}
97 1+(7.49e5+1.29e6i)T+(−4.03e13+6.99e13i)T2 1 + (7.49e5 + 1.29e6i)T + (-4.03e13 + 6.99e13i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.56404872717863128056281696007, −12.59439585237252791583954770339, −11.15568342412260940664651502224, −10.79590861746968249347117923903, −9.373569653494822376855036557367, −7.21280759177989262647373059046, −6.57415682665465319637395404124, −5.14977954370635497917892975723, −3.44227940775691191010931608812, −2.20057912499640660002424996681, 0.10080551964344067204724181447, 2.14296355369613909411422223393, 4.16891873772899023385634132624, 5.39460229673568699128687260589, 6.09074378641107511274378034961, 8.200429153763231642361137243270, 8.883796006681935556059466862694, 10.71564429617088268867872814741, 11.66282989445871133556455387790, 12.99882588239031588581662061439

Graph of the ZZ-function along the critical line