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Properties

Label 2-435-145.144-c1-0-12
Degree $2$
Conductor $435$
Sign $0.448 + 0.893i$
Analytic cond. $3.47349$
Root an. cond. $1.86373$
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  − 1.97·2-s − 3-s + 1.89·4-s + (1.38 − 1.75i)5-s + 1.97·6-s + 0.520i·7-s + 0.214·8-s + 9-s + (−2.72 + 3.46i)10-s − 2.16i·11-s − 1.89·12-s + 1.53i·13-s − 1.02i·14-s + (−1.38 + 1.75i)15-s − 4.20·16-s + 2.47·17-s + ⋯
L(s)  = 1  − 1.39·2-s − 0.577·3-s + 0.945·4-s + (0.617 − 0.786i)5-s + 0.805·6-s + 0.196i·7-s + 0.0758·8-s + 0.333·9-s + (−0.861 + 1.09i)10-s − 0.652i·11-s − 0.545·12-s + 0.425i·13-s − 0.274i·14-s + (−0.356 + 0.454i)15-s − 1.05·16-s + 0.601·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.448 + 0.893i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ 0.448 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.501384 - 0.309462i\)
\(L(\frac12)\) \(\approx\) \(0.501384 - 0.309462i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + (-1.38 + 1.75i)T \)
29 \( 1 + (-2.29 + 4.87i)T \)
good2 \( 1 + 1.97T + 2T^{2} \)
7 \( 1 - 0.520iT - 7T^{2} \)
11 \( 1 + 2.16iT - 11T^{2} \)
13 \( 1 - 1.53iT - 13T^{2} \)
17 \( 1 - 2.47T + 17T^{2} \)
19 \( 1 - 4.70iT - 19T^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
31 \( 1 + 4.64iT - 31T^{2} \)
37 \( 1 - 1.77T + 37T^{2} \)
41 \( 1 + 9.71iT - 41T^{2} \)
43 \( 1 - 5.39T + 43T^{2} \)
47 \( 1 - 3.68T + 47T^{2} \)
53 \( 1 + 7.74iT - 53T^{2} \)
59 \( 1 + 1.91T + 59T^{2} \)
61 \( 1 + 7.66iT - 61T^{2} \)
67 \( 1 + 7.88iT - 67T^{2} \)
71 \( 1 + 9.78T + 71T^{2} \)
73 \( 1 + 4.70T + 73T^{2} \)
79 \( 1 - 12.3iT - 79T^{2} \)
83 \( 1 + 7.32iT - 83T^{2} \)
89 \( 1 - 7.66iT - 89T^{2} \)
97 \( 1 + 1.64T + 97T^{2} \)
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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

−10.67305537593214080853953781178, −9.963177360252382493408876557808, −9.227828649709376807035745704167, −8.422400883984321005602303818435, −7.60591039136979406012883783845, −6.30520536078259477546922755290, −5.50199111514738442427427111391, −4.19185150448646007936721143228, −2.05689704942115635023617649873, −0.74035514304204564601057686220, 1.28314545827527783757071887383, 2.79363568961288372718918188206, 4.63372524825557059587573872200, 5.88797078196181595215712003270, 7.03758658382554884502711087736, 7.46136687014584946002846181141, 8.786248311154622492071496998094, 9.636508696064823728799599702272, 10.36244088380221644216095378385, 10.84229579715468130348142278173

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