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Properties

Label 2-357-357.356-c1-0-4
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

\(L(1)\) \(\approx\) \(0.576101 - 0.990088i\)
\(L(\frac12)\) \(\approx\) \(0.576101 - 0.990088i\)
\(L(\frac{3}{2})\) 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

−12.81330533470392055596486925536, −10.58989737400183710530722295639, −9.781739179206582961673939362288, −9.060131894632347144755241560423, −8.195862891276581157743742685073, −7.27087665230912954002905359066, −6.53553125886644087875001035998, −5.51664495814045845800230600192, −4.18138398863387878764011647887, −3.31058786248129787303646870864, 0.75062837618358383043958419588, 2.23987195323024870131040670703, 3.25639888151414111081974689714, 4.19393869309727350084619295730, 5.50339956774371113952966096425, 7.38113138585060618486783113665, 8.703889019598361302805316791400, 9.153163995165514228288489308525, 9.837127789587178040791512181343, 10.86612299216378179677419423632

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