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Properties

Label 2-74-37.26-c7-0-4
Degree 22
Conductor 7474
Sign −0.727+0.685i-0.727 + 0.685i
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  + (4 + 6.92i)2-s + (1.38 − 2.39i)3-s + (−31.9 + 55.4i)4-s + (−125. + 216. i)5-s + 22.1·6-s + (133. − 231. i)7-s − 511.·8-s + (1.08e3 + 1.88e3i)9-s − 2.00e3·10-s − 2.17e3·11-s + (88.5 + 153. i)12-s + (−2.08e3 + 3.60e3i)13-s + 2.13e3·14-s + (345. + 598. i)15-s + (−2.04e3 − 3.54e3i)16-s + (−1.80e4 − 3.12e4i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.0295 − 0.0512i)3-s + (−0.249 + 0.433i)4-s + (−0.447 + 0.774i)5-s + 0.0418·6-s + (0.147 − 0.254i)7-s − 0.353·8-s + (0.498 + 0.862i)9-s − 0.632·10-s − 0.492·11-s + (0.0147 + 0.0256i)12-s + (−0.262 + 0.454i)13-s + 0.207·14-s + (0.0264 + 0.0458i)15-s + (−0.125 − 0.216i)16-s + (−0.889 − 1.54i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((−0.727+0.685i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=((−0.727+0.685i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: −0.727+0.685i-0.727 + 0.685i
Analytic conductor: 23.116423.1164
Root analytic conductor: 4.807964.80796
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ74(63,⋅)\chi_{74} (63, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 74, ( :7/2), −0.727+0.685i)(2,\ 74,\ (\ :7/2),\ -0.727 + 0.685i)

Particular Values

L(4)L(4) ≈\approx 0.199275−0.502019i0.199275 - 0.502019i
L(12)L(\frac12) ≈\approx 0.199275−0.502019i0.199275 - 0.502019i
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+(−4−6.92i)T 1 + (-4 - 6.92i)T
37 1+(−1.90e5+2.41e5i)T 1 + (-1.90e5 + 2.41e5i)T
good3 1+(−1.38+2.39i)T+(−1.09e3−1.89e3i)T2 1 + (-1.38 + 2.39i)T + (-1.09e3 - 1.89e3i)T^{2}
5 1+(125.−216.i)T+(−3.90e4−6.76e4i)T2 1 + (125. - 216. i)T + (-3.90e4 - 6.76e4i)T^{2}
7 1+(−133.+231.i)T+(−4.11e5−7.13e5i)T2 1 + (-133. + 231. i)T + (-4.11e5 - 7.13e5i)T^{2}
11 1+2.17e3T+1.94e7T2 1 + 2.17e3T + 1.94e7T^{2}
13 1+(2.08e3−3.60e3i)T+(−3.13e7−5.43e7i)T2 1 + (2.08e3 - 3.60e3i)T + (-3.13e7 - 5.43e7i)T^{2}
17 1+(1.80e4+3.12e4i)T+(−2.05e8+3.55e8i)T2 1 + (1.80e4 + 3.12e4i)T + (-2.05e8 + 3.55e8i)T^{2}
19 1+(−2.33e3+4.04e3i)T+(−4.46e8−7.74e8i)T2 1 + (-2.33e3 + 4.04e3i)T + (-4.46e8 - 7.74e8i)T^{2}
23 1+4.00e4T+3.40e9T2 1 + 4.00e4T + 3.40e9T^{2}
29 1+8.54e4T+1.72e10T2 1 + 8.54e4T + 1.72e10T^{2}
31 1+3.19e5T+2.75e10T2 1 + 3.19e5T + 2.75e10T^{2}
41 1+(−3.95e5+6.85e5i)T+(−9.73e10−1.68e11i)T2 1 + (-3.95e5 + 6.85e5i)T + (-9.73e10 - 1.68e11i)T^{2}
43 1+7.88e5T+2.71e11T2 1 + 7.88e5T + 2.71e11T^{2}
47 1−1.31e5T+5.06e11T2 1 - 1.31e5T + 5.06e11T^{2}
53 1+(4.24e5+7.35e5i)T+(−5.87e11+1.01e12i)T2 1 + (4.24e5 + 7.35e5i)T + (-5.87e11 + 1.01e12i)T^{2}
59 1+(−1.34e6−2.33e6i)T+(−1.24e12+2.15e12i)T2 1 + (-1.34e6 - 2.33e6i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(1.10e6−1.92e6i)T+(−1.57e12−2.72e12i)T2 1 + (1.10e6 - 1.92e6i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(8.67e5−1.50e6i)T+(−3.03e12−5.24e12i)T2 1 + (8.67e5 - 1.50e6i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1+(1.24e6−2.15e6i)T+(−4.54e12−7.87e12i)T2 1 + (1.24e6 - 2.15e6i)T + (-4.54e12 - 7.87e12i)T^{2}
73 1+1.24e6T+1.10e13T2 1 + 1.24e6T + 1.10e13T^{2}
79 1+(−1.61e5+2.79e5i)T+(−9.60e12−1.66e13i)T2 1 + (-1.61e5 + 2.79e5i)T + (-9.60e12 - 1.66e13i)T^{2}
83 1+(−3.84e6−6.65e6i)T+(−1.35e13+2.35e13i)T2 1 + (-3.84e6 - 6.65e6i)T + (-1.35e13 + 2.35e13i)T^{2}
89 1+(−2.91e6−5.04e6i)T+(−2.21e13+3.83e13i)T2 1 + (-2.91e6 - 5.04e6i)T + (-2.21e13 + 3.83e13i)T^{2}
97 1+3.96e6T+8.07e13T2 1 + 3.96e6T + 8.07e13T^{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

−13.88405187321213974318180652866, −13.01528228020440442727731596539, −11.55950908245911827538953259127, −10.63802223163625687621051272287, −9.145631124005076270975637464916, −7.50294409241512277093089617711, −7.12657980350889957547968966009, −5.33600003721008846113635268655, −4.06689913531336768521233059857, −2.40371313111655631005759246952, 0.15657865428901898624854667646, 1.73887517844640876693006621218, 3.60407246025310478302577254893, 4.73758940595289300230282958458, 6.17034460147888835756436761120, 7.980899548586941063270744310339, 9.102069841257803144709140885167, 10.29161988789305595076531778872, 11.50076348956810480159564326401, 12.64729452655201518970918675233

Graph of the ZZ-function along the critical line