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Properties

Label 2-357-17.13-c1-0-7
Degree 22
Conductor 357357
Sign 0.292−0.956i0.292 - 0.956i
Analytic cond. 2.850652.85065
Root an. cond. 1.688381.68838
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  + 0.901i·2-s + (0.707 + 0.707i)3-s + 1.18·4-s + (−0.446 − 0.446i)5-s + (−0.637 + 0.637i)6-s + (0.707 − 0.707i)7-s + 2.87i·8-s + 1.00i·9-s + (0.402 − 0.402i)10-s + (−0.261 + 0.261i)11-s + (0.839 + 0.839i)12-s + 4.13·13-s + (0.637 + 0.637i)14-s − 0.631i·15-s − 0.217·16-s + (−3.84 + 1.47i)17-s + ⋯
L(s)  = 1  + 0.637i·2-s + (0.408 + 0.408i)3-s + 0.593·4-s + (−0.199 − 0.199i)5-s + (−0.260 + 0.260i)6-s + (0.267 − 0.267i)7-s + 1.01i·8-s + 0.333i·9-s + (0.127 − 0.127i)10-s + (−0.0787 + 0.0787i)11-s + (0.242 + 0.242i)12-s + 1.14·13-s + (0.170 + 0.170i)14-s − 0.162i·15-s − 0.0544·16-s + (−0.933 + 0.358i)17-s + ⋯

Functional equation

Λ(s)=(357s/2 ΓC(s) L(s)=((0.292−0.956i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(357s/2 ΓC(s+1/2) L(s)=((0.292−0.956i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 357357    =    3⋅7⋅173 \cdot 7 \cdot 17
Sign: 0.292−0.956i0.292 - 0.956i
Analytic conductor: 2.850652.85065
Root analytic conductor: 1.688381.68838
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ357(64,⋅)\chi_{357} (64, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 357, ( :1/2), 0.292−0.956i)(2,\ 357,\ (\ :1/2),\ 0.292 - 0.956i)

Particular Values

L(1)L(1) ≈\approx 1.44384+1.06847i1.44384 + 1.06847i
L(12)L(\frac12) ≈\approx 1.44384+1.06847i1.44384 + 1.06847i
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)
bad3 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
7 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
17 1+(3.84−1.47i)T 1 + (3.84 - 1.47i)T
good2 1−0.901iT−2T2 1 - 0.901iT - 2T^{2}
5 1+(0.446+0.446i)T+5iT2 1 + (0.446 + 0.446i)T + 5iT^{2}
11 1+(0.261−0.261i)T−11iT2 1 + (0.261 - 0.261i)T - 11iT^{2}
13 1−4.13T+13T2 1 - 4.13T + 13T^{2}
19 1−5.75iT−19T2 1 - 5.75iT - 19T^{2}
23 1+(−4.55+4.55i)T−23iT2 1 + (-4.55 + 4.55i)T - 23iT^{2}
29 1+(0.469+0.469i)T+29iT2 1 + (0.469 + 0.469i)T + 29iT^{2}
31 1+(6.86+6.86i)T+31iT2 1 + (6.86 + 6.86i)T + 31iT^{2}
37 1+(−0.683−0.683i)T+37iT2 1 + (-0.683 - 0.683i)T + 37iT^{2}
41 1+(4.61−4.61i)T−41iT2 1 + (4.61 - 4.61i)T - 41iT^{2}
43 1−4.21iT−43T2 1 - 4.21iT - 43T^{2}
47 1+8.03T+47T2 1 + 8.03T + 47T^{2}
53 1+7.76iT−53T2 1 + 7.76iT - 53T^{2}
59 1+8.34iT−59T2 1 + 8.34iT - 59T^{2}
61 1+(−8.97+8.97i)T−61iT2 1 + (-8.97 + 8.97i)T - 61iT^{2}
67 1−3.38T+67T2 1 - 3.38T + 67T^{2}
71 1+(−6.55−6.55i)T+71iT2 1 + (-6.55 - 6.55i)T + 71iT^{2}
73 1+(9.51+9.51i)T+73iT2 1 + (9.51 + 9.51i)T + 73iT^{2}
79 1+(−3.19+3.19i)T−79iT2 1 + (-3.19 + 3.19i)T - 79iT^{2}
83 1+0.439iT−83T2 1 + 0.439iT - 83T^{2}
89 1−2.28T+89T2 1 - 2.28T + 89T^{2}
97 1+(3.02+3.02i)T+97iT2 1 + (3.02 + 3.02i)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

−11.32619668405970750129569386411, −10.88993840462518503592383815540, −9.776369754124909374525486438337, −8.391349495049556032312260015563, −8.156278913124935943352536583086, −6.83484628214683100497859821355, −5.99181742071967765521849810797, −4.72436575078842120063985150208, −3.54582019092450176361340903814, −1.96712149158375338958647191287, 1.45142440448376912661360516778, 2.76176766532891929614882129565, 3.72665515405278484084301949742, 5.38151449312909546794528360519, 6.78302271983695354509791336191, 7.27240585077369719164685386588, 8.670139326874904726971320411764, 9.311667182281476638821361677432, 10.80440885292246356644903138532, 11.16505870162185863004745988655

Graph of the ZZ-function along the critical line