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Properties

Label 2-357-17.13-c1-0-7
Degree $2$
Conductor $357$
Sign $0.292 - 0.956i$
Analytic cond. $2.85065$
Root an. cond. $1.68838$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.292 - 0.956i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ 0.292 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44384 + 1.06847i\)
\(L(\frac12)\) \(\approx\) \(1.44384 + 1.06847i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (3.84 - 1.47i)T \)
good2 \( 1 - 0.901iT - 2T^{2} \)
5 \( 1 + (0.446 + 0.446i)T + 5iT^{2} \)
11 \( 1 + (0.261 - 0.261i)T - 11iT^{2} \)
13 \( 1 - 4.13T + 13T^{2} \)
19 \( 1 - 5.75iT - 19T^{2} \)
23 \( 1 + (-4.55 + 4.55i)T - 23iT^{2} \)
29 \( 1 + (0.469 + 0.469i)T + 29iT^{2} \)
31 \( 1 + (6.86 + 6.86i)T + 31iT^{2} \)
37 \( 1 + (-0.683 - 0.683i)T + 37iT^{2} \)
41 \( 1 + (4.61 - 4.61i)T - 41iT^{2} \)
43 \( 1 - 4.21iT - 43T^{2} \)
47 \( 1 + 8.03T + 47T^{2} \)
53 \( 1 + 7.76iT - 53T^{2} \)
59 \( 1 + 8.34iT - 59T^{2} \)
61 \( 1 + (-8.97 + 8.97i)T - 61iT^{2} \)
67 \( 1 - 3.38T + 67T^{2} \)
71 \( 1 + (-6.55 - 6.55i)T + 71iT^{2} \)
73 \( 1 + (9.51 + 9.51i)T + 73iT^{2} \)
79 \( 1 + (-3.19 + 3.19i)T - 79iT^{2} \)
83 \( 1 + 0.439iT - 83T^{2} \)
89 \( 1 - 2.28T + 89T^{2} \)
97 \( 1 + (3.02 + 3.02i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line