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Properties

Label 2-357-357.356-c1-0-6
Degree $2$
Conductor $357$
Sign $0.494 + 0.869i$
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  − 2.82i·2-s + (−1.50 + 0.855i)3-s − 5.97·4-s + 1.62i·5-s + (2.41 + 4.25i)6-s + 2.64·7-s + 11.2i·8-s + (1.53 − 2.57i)9-s + 4.58·10-s + (9.00 − 5.11i)12-s − 7.47i·14-s + (−1.39 − 2.44i)15-s + 19.7·16-s + 4.12i·17-s + (−7.27 − 4.33i)18-s + ⋯
L(s)  = 1  − 1.99i·2-s + (−0.869 + 0.494i)3-s − 2.98·4-s + 0.726i·5-s + (0.986 + 1.73i)6-s + 0.999·7-s + 3.97i·8-s + (0.511 − 0.859i)9-s + 1.45·10-s + (2.59 − 1.47i)12-s − 1.99i·14-s + (−0.358 − 0.631i)15-s + 4.94·16-s + 0.999i·17-s + (−1.71 − 1.02i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.494 + 0.869i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (356, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ 0.494 + 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.747963 - 0.435215i\)
\(L(\frac12)\) \(\approx\) \(0.747963 - 0.435215i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.50 - 0.855i)T \)
7 \( 1 - 2.64T \)
17 \( 1 - 4.12iT \)
good2 \( 1 + 2.82iT - 2T^{2} \)
5 \( 1 - 1.62iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 8.54T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 9.80iT - 41T^{2} \)
43 \( 1 - 8.27T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6.86iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 + 1.25T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 3.81T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 4.58T + 97T^{2} \)
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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

−11.11726070955975259361622261436, −10.69546126252823868430638252386, −10.00475384515308934393686517560, −8.978668773804484456873143040204, −7.921563350805488326600384952446, −6.03867659103321671637909870324, −4.84188654300570774455565523241, −4.12088034240296247269251051747, −2.84620308782396722512887318886, −1.30515950029661245944993865065, 0.841743919941709298309676730328, 4.42942331606928429436341910788, 5.01742713764280495084829809263, 5.80566475532984920500047323441, 6.87906952217544812451245082489, 7.65494675396607624937837491827, 8.421717365571444988686378652625, 9.316778974832353887993410387320, 10.52385154023668801862742203764, 11.94257753790783286097570817232

Graph of the $Z$-function along the critical line