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Properties

Label 2-357-21.20-c1-0-6
Degree $2$
Conductor $357$
Sign $-0.985 + 0.172i$
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  + 1.22i·2-s + (0.136 + 1.72i)3-s + 0.490·4-s − 3.00·5-s + (−2.12 + 0.167i)6-s + (0.249 + 2.63i)7-s + 3.05i·8-s + (−2.96 + 0.470i)9-s − 3.69i·10-s − 2.71i·11-s + (0.0668 + 0.846i)12-s − 2.99i·13-s + (−3.23 + 0.306i)14-s + (−0.409 − 5.18i)15-s − 2.77·16-s + 17-s + ⋯
L(s)  = 1  + 0.868i·2-s + (0.0787 + 0.996i)3-s + 0.245·4-s − 1.34·5-s + (−0.866 + 0.0684i)6-s + (0.0944 + 0.995i)7-s + 1.08i·8-s + (−0.987 + 0.156i)9-s − 1.16i·10-s − 0.817i·11-s + (0.0193 + 0.244i)12-s − 0.830i·13-s + (−0.864 + 0.0820i)14-s + (−0.105 − 1.33i)15-s − 0.694·16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $-0.985 + 0.172i$
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: \((2,\ 357,\ (\ :1/2),\ -0.985 + 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0903066 - 1.03919i\)
\(L(\frac12)\) \(\approx\) \(0.0903066 - 1.03919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.136 - 1.72i)T \)
7 \( 1 + (-0.249 - 2.63i)T \)
17 \( 1 - T \)
good2 \( 1 - 1.22iT - 2T^{2} \)
5 \( 1 + 3.00T + 5T^{2} \)
11 \( 1 + 2.71iT - 11T^{2} \)
13 \( 1 + 2.99iT - 13T^{2} \)
19 \( 1 - 6.46iT - 19T^{2} \)
23 \( 1 - 0.531iT - 23T^{2} \)
29 \( 1 - 8.12iT - 29T^{2} \)
31 \( 1 + 4.76iT - 31T^{2} \)
37 \( 1 - 0.597T + 37T^{2} \)
41 \( 1 + 4.72T + 41T^{2} \)
43 \( 1 - 8.79T + 43T^{2} \)
47 \( 1 - 5.20T + 47T^{2} \)
53 \( 1 - 1.86iT - 53T^{2} \)
59 \( 1 - 14.9T + 59T^{2} \)
61 \( 1 - 8.34iT - 61T^{2} \)
67 \( 1 - 6.91T + 67T^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 - 4.26iT - 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 + 3.09T + 83T^{2} \)
89 \( 1 - 9.36T + 89T^{2} \)
97 \( 1 + 5.63iT - 97T^{2} \)
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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

−11.68939492226865921915071140951, −11.16017573310516414660547517693, −10.16249612960593301610282135526, −8.681219398511096554265826410392, −8.305974302594526315340349342039, −7.41717084005351193550555319884, −5.87808994218932171058678176064, −5.37795286255109886932526700893, −3.90909210861899381004210830069, −2.86864972873841162550747490133, 0.70191263744714947815682674104, 2.24106003454082151270533782981, 3.58473671400073277282828090309, 4.54803525005967410557135476913, 6.62624427194283328400909567727, 7.19507194314205932519304663991, 7.87617425067045282977335479998, 9.166584022815000983132878427361, 10.37977534675787046522221050409, 11.39056463091312888708480935297

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