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Properties

Label 2-3150-15.8-c1-0-13
Degree 22
Conductor 31503150
Sign 0.999+0.0387i0.999 + 0.0387i
Analytic cond. 25.152825.1528
Root an. cond. 5.015265.01526
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + 0.585i·11-s + (−3.43 − 3.43i)13-s + 1.00·14-s − 1.00·16-s + (0.906 + 0.906i)17-s + 2.04i·19-s + (0.414 − 0.414i)22-s + (−0.257 + 0.257i)23-s + 4.86i·26-s + (−0.707 − 0.707i)28-s + 1.75·29-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + 0.176i·11-s + (−0.953 − 0.953i)13-s + 0.267·14-s − 0.250·16-s + (0.219 + 0.219i)17-s + 0.470i·19-s + (0.0883 − 0.0883i)22-s + (−0.0537 + 0.0537i)23-s + 0.953i·26-s + (−0.133 − 0.133i)28-s + 0.325·29-s + ⋯

Functional equation

Λ(s)=(3150s/2 ΓC(s) L(s)=((0.999+0.0387i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0387i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3150s/2 ΓC(s+1/2) L(s)=((0.999+0.0387i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31503150    =    2⋅32⋅52⋅72 \cdot 3^{2} \cdot 5^{2} \cdot 7
Sign: 0.999+0.0387i0.999 + 0.0387i
Analytic conductor: 25.152825.1528
Root analytic conductor: 5.015265.01526
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3150(2843,⋅)\chi_{3150} (2843, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3150, ( :1/2), 0.999+0.0387i)(2,\ 3150,\ (\ :1/2),\ 0.999 + 0.0387i)

Particular Values

L(1)L(1) ≈\approx 1.1215642431.121564243
L(12)L(\frac12) ≈\approx 1.1215642431.121564243
L(32)L(\frac{3}{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+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
3 1 1
5 1 1
7 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good11 1−0.585iT−11T2 1 - 0.585iT - 11T^{2}
13 1+(3.43+3.43i)T+13iT2 1 + (3.43 + 3.43i)T + 13iT^{2}
17 1+(−0.906−0.906i)T+17iT2 1 + (-0.906 - 0.906i)T + 17iT^{2}
19 1−2.04iT−19T2 1 - 2.04iT - 19T^{2}
23 1+(0.257−0.257i)T−23iT2 1 + (0.257 - 0.257i)T - 23iT^{2}
29 1−1.75T+29T2 1 - 1.75T + 29T^{2}
31 1−1.47T+31T2 1 - 1.47T + 31T^{2}
37 1+(−4.04+4.04i)T−37iT2 1 + (-4.04 + 4.04i)T - 37iT^{2}
41 1−3.84iT−41T2 1 - 3.84iT - 41T^{2}
43 1+(5.10+5.10i)T+43iT2 1 + (5.10 + 5.10i)T + 43iT^{2}
47 1+(−5.49−5.49i)T+47iT2 1 + (-5.49 - 5.49i)T + 47iT^{2}
53 1+(5.02−5.02i)T−53iT2 1 + (5.02 - 5.02i)T - 53iT^{2}
59 1−11.1T+59T2 1 - 11.1T + 59T^{2}
61 1−3.78T+61T2 1 - 3.78T + 61T^{2}
67 1+(−6.74+6.74i)T−67iT2 1 + (-6.74 + 6.74i)T - 67iT^{2}
71 1−10.9iT−71T2 1 - 10.9iT - 71T^{2}
73 1+(5.97+5.97i)T+73iT2 1 + (5.97 + 5.97i)T + 73iT^{2}
79 1−0.944iT−79T2 1 - 0.944iT - 79T^{2}
83 1+(−7.82+7.82i)T−83iT2 1 + (-7.82 + 7.82i)T - 83iT^{2}
89 1−0.0705T+89T2 1 - 0.0705T + 89T^{2}
97 1+(−0.746+0.746i)T−97iT2 1 + (-0.746 + 0.746i)T - 97iT^{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

−8.741182750857460192288702023368, −7.941164031535419761884902130393, −7.44822949888239034101951556581, −6.47995063762519355083123325795, −5.62125663045490802222096777189, −4.78961337563564726228340486511, −3.77188498422400026478529536334, −2.89598299012985863014463256416, −2.10758038220133630552309515733, −0.77592535775976521421220301103, 0.59527336924763467924883553242, 1.94637393172584072472016517687, 2.96360891159280893940080744495, 4.15777589451826408215959423702, 4.90448563380761616764360616930, 5.73799741330026210888298053556, 6.79460264218925820319301395069, 6.98016261284486714441154568674, 7.988552528446080432047316521128, 8.607939050287288016438270917200

Graph of the ZZ-function along the critical line