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Properties

Label 2-70-35.18-c2-0-0
Degree 22
Conductor 7070
Sign −0.644−0.764i-0.644 - 0.764i
Analytic cond. 1.907361.90736
Root an. cond. 1.381071.38107
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.366i)2-s + (−5.10 + 1.36i)3-s + (1.73 + i)4-s + (−1.96 + 4.59i)5-s − 7.47·6-s + (−6.41 + 2.80i)7-s + (1.99 + 2i)8-s + (16.4 − 9.47i)9-s + (−4.36 + 5.56i)10-s + (−1.73 + 3.01i)11-s + (−10.2 − 2.73i)12-s + (10.9 + 10.9i)13-s + (−9.78 + 1.47i)14-s + (3.73 − 26.1i)15-s + (1.99 + 3.46i)16-s + (−2.54 − 9.49i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−1.70 + 0.456i)3-s + (0.433 + 0.250i)4-s + (−0.392 + 0.919i)5-s − 1.24·6-s + (−0.916 + 0.400i)7-s + (0.249 + 0.250i)8-s + (1.82 − 1.05i)9-s + (−0.436 + 0.556i)10-s + (−0.158 + 0.273i)11-s + (−0.851 − 0.228i)12-s + (0.842 + 0.842i)13-s + (−0.699 + 0.105i)14-s + (0.249 − 1.74i)15-s + (0.124 + 0.216i)16-s + (−0.149 − 0.558i)17-s + ⋯

Functional equation

Λ(s)=(70s/2 ΓC(s) L(s)=((−0.644−0.764i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(70s/2 ΓC(s+1) L(s)=((−0.644−0.764i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7070    =    2⋅5⋅72 \cdot 5 \cdot 7
Sign: −0.644−0.764i-0.644 - 0.764i
Analytic conductor: 1.907361.90736
Root analytic conductor: 1.381071.38107
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ70(53,⋅)\chi_{70} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 70, ( :1), −0.644−0.764i)(2,\ 70,\ (\ :1),\ -0.644 - 0.764i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.346080+0.743919i0.346080 + 0.743919i
L(12)L(\frac12) ≈\approx 0.346080+0.743919i0.346080 + 0.743919i
L(2)L(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+(−1.36−0.366i)T 1 + (-1.36 - 0.366i)T
5 1+(1.96−4.59i)T 1 + (1.96 - 4.59i)T
7 1+(6.41−2.80i)T 1 + (6.41 - 2.80i)T
good3 1+(5.10−1.36i)T+(7.79−4.5i)T2 1 + (5.10 - 1.36i)T + (7.79 - 4.5i)T^{2}
11 1+(1.73−3.01i)T+(−60.5−104.i)T2 1 + (1.73 - 3.01i)T + (-60.5 - 104. i)T^{2}
13 1+(−10.9−10.9i)T+169iT2 1 + (-10.9 - 10.9i)T + 169iT^{2}
17 1+(2.54+9.49i)T+(−250.+144.5i)T2 1 + (2.54 + 9.49i)T + (-250. + 144.5i)T^{2}
19 1+(−15.9+9.21i)T+(180.5−312.i)T2 1 + (-15.9 + 9.21i)T + (180.5 - 312. i)T^{2}
23 1+(4.64−17.3i)T+(−458.−264.5i)T2 1 + (4.64 - 17.3i)T + (-458. - 264.5i)T^{2}
29 1−49.8iT−841T2 1 - 49.8iT - 841T^{2}
31 1+(8−13.8i)T+(−480.5−832.i)T2 1 + (8 - 13.8i)T + (-480.5 - 832. i)T^{2}
37 1+(−34.2−9.16i)T+(1.18e3+684.5i)T2 1 + (-34.2 - 9.16i)T + (1.18e3 + 684.5i)T^{2}
41 1+6.04T+1.68e3T2 1 + 6.04T + 1.68e3T^{2}
43 1+(9.64+9.64i)T+1.84e3iT2 1 + (9.64 + 9.64i)T + 1.84e3iT^{2}
47 1+(20.3+5.44i)T+(1.91e3+1.10e3i)T2 1 + (20.3 + 5.44i)T + (1.91e3 + 1.10e3i)T^{2}
53 1+(−10.9+2.94i)T+(2.43e3−1.40e3i)T2 1 + (-10.9 + 2.94i)T + (2.43e3 - 1.40e3i)T^{2}
59 1+(−65.3−37.7i)T+(1.74e3+3.01e3i)T2 1 + (-65.3 - 37.7i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(51.4+89.1i)T+(−1.86e3+3.22e3i)T2 1 + (51.4 + 89.1i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(8.33+31.1i)T+(−3.88e3+2.24e3i)T2 1 + (8.33 + 31.1i)T + (-3.88e3 + 2.24e3i)T^{2}
71 1−40.3T+5.04e3T2 1 - 40.3T + 5.04e3T^{2}
73 1+(−85.7+22.9i)T+(4.61e3−2.66e3i)T2 1 + (-85.7 + 22.9i)T + (4.61e3 - 2.66e3i)T^{2}
79 1+(−87.6+50.6i)T+(3.12e3−5.40e3i)T2 1 + (-87.6 + 50.6i)T + (3.12e3 - 5.40e3i)T^{2}
83 1+(−51.1−51.1i)T+6.88e3iT2 1 + (-51.1 - 51.1i)T + 6.88e3iT^{2}
89 1+(71.6−41.3i)T+(3.96e3−6.85e3i)T2 1 + (71.6 - 41.3i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(−45.7+45.7i)T−9.40e3iT2 1 + (-45.7 + 45.7i)T - 9.40e3iT^{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

−15.18085707329436084179514678990, −13.69045125919544770859895652127, −12.38466225653843643190713837271, −11.54516519336027384095024019374, −10.81720255611793575213223692983, −9.550253823048233940851589460999, −7.05431888732550479459770588833, −6.34668454974522843695412420526, −5.12132010678239212373124961946, −3.55668901443372245323589294887, 0.73756352145498310504947208972, 4.06551705740924862542412377917, 5.54202476405790014026715186385, 6.31720730031604419424802306126, 7.84694677599315320243156362139, 9.981597865301497605769947742289, 11.06880734278230096222867378376, 12.00753600113023323191495236438, 12.88605638794226141721781754304, 13.40535885205262125836358771150

Graph of the ZZ-function along the critical line