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Properties

Label 2-357-21.20-c1-0-15
Degree $2$
Conductor $357$
Sign $-0.0561 - 0.998i$
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  + 0.886i·2-s + (0.730 + 1.57i)3-s + 1.21·4-s + 2.83·5-s + (−1.39 + 0.647i)6-s + (−2.45 − 0.979i)7-s + 2.84i·8-s + (−1.93 + 2.29i)9-s + 2.51i·10-s + 0.938i·11-s + (0.887 + 1.90i)12-s − 3.81i·13-s + (0.868 − 2.17i)14-s + (2.07 + 4.45i)15-s − 0.0953·16-s + 17-s + ⋯
L(s)  = 1  + 0.626i·2-s + (0.421 + 0.906i)3-s + 0.607·4-s + 1.26·5-s + (−0.568 + 0.264i)6-s + (−0.928 − 0.370i)7-s + 1.00i·8-s + (−0.644 + 0.764i)9-s + 0.794i·10-s + 0.283i·11-s + (0.256 + 0.550i)12-s − 1.05i·13-s + (0.232 − 0.582i)14-s + (0.534 + 1.14i)15-s − 0.0238·16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0561 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0561 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $-0.0561 - 0.998i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ -0.0561 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33831 + 1.41564i\)
\(L(\frac12)\) \(\approx\) \(1.33831 + 1.41564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.730 - 1.57i)T \)
7 \( 1 + (2.45 + 0.979i)T \)
17 \( 1 - T \)
good2 \( 1 - 0.886iT - 2T^{2} \)
5 \( 1 - 2.83T + 5T^{2} \)
11 \( 1 - 0.938iT - 11T^{2} \)
13 \( 1 + 3.81iT - 13T^{2} \)
19 \( 1 + 3.64iT - 19T^{2} \)
23 \( 1 - 1.96iT - 23T^{2} \)
29 \( 1 + 5.06iT - 29T^{2} \)
31 \( 1 - 1.71iT - 31T^{2} \)
37 \( 1 + 5.82T + 37T^{2} \)
41 \( 1 - 4.40T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 3.33T + 47T^{2} \)
53 \( 1 + 8.71iT - 53T^{2} \)
59 \( 1 + 0.484T + 59T^{2} \)
61 \( 1 - 0.444iT - 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + 15.8iT - 73T^{2} \)
79 \( 1 - 7.20T + 79T^{2} \)
83 \( 1 + 7.26T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 - 9.09iT - 97T^{2} \)
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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

−11.46366622669007043813668962834, −10.36817965475391545696721452895, −9.980048579342602165181217312890, −9.030805151407163278716272230759, −7.916654715098891931925890437936, −6.81071878198822451968743944289, −5.85589817717474680752091334522, −5.08971429348553570743383149259, −3.35910761977983321118346645839, −2.31318934652540468888762866084, 1.56030943062358038586156378460, 2.46924529853089538980154760914, 3.52763082581094643674832645865, 5.75146638369816166938846430910, 6.43077877310366881534455053572, 7.15391561801809316683050290749, 8.637725180830295917056379049184, 9.515009474483553890238747743637, 10.18761544500953507587915414868, 11.36044081936324240227996282100

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