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Properties

Label 2-468-156.59-c0-0-0
Degree $2$
Conductor $468$
Sign $-0.918 - 0.394i$
Analytic cond. $0.233562$
Root an. cond. $0.483282$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−1.22 + 1.22i)5-s + (−0.707 − 0.707i)8-s + (−1.49 − 0.866i)10-s + (−0.5 + 0.866i)13-s + (0.500 − 0.866i)16-s + (0.258 + 0.448i)17-s + (0.448 − 1.67i)20-s − 1.99i·25-s + (−0.965 − 0.258i)26-s + (1.67 + 0.965i)29-s + (0.965 + 0.258i)32-s + (−0.366 + 0.366i)34-s + (0.5 + 1.86i)37-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−1.22 + 1.22i)5-s + (−0.707 − 0.707i)8-s + (−1.49 − 0.866i)10-s + (−0.5 + 0.866i)13-s + (0.500 − 0.866i)16-s + (0.258 + 0.448i)17-s + (0.448 − 1.67i)20-s − 1.99i·25-s + (−0.965 − 0.258i)26-s + (1.67 + 0.965i)29-s + (0.965 + 0.258i)32-s + (−0.366 + 0.366i)34-s + (0.5 + 1.86i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $-0.918 - 0.394i$
Analytic conductor: \(0.233562\)
Root analytic conductor: \(0.483282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :0),\ -0.918 - 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6635764196\)
\(L(\frac12)\) \(\approx\) \(0.6635764196\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + 1.93iT - T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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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

−11.83856684594369918621598599477, −10.75412016901591413951219253664, −9.823493494584027824941221059513, −8.567543361446601179439289912423, −7.85492557032490232628541188985, −6.91557887759875743327631571435, −6.47434842501836238853343755249, −4.93257593728763114415825579104, −3.95194803535709756617643671174, −2.98191306128880075412118192744, 0.841174426196479551750983974367, 2.79188106071294506308134056415, 4.06093059798509731247237750447, 4.77175819767595790785267465039, 5.74156829892804597679595121412, 7.53618739665680960246933781059, 8.306928634603320384466526805222, 9.118541237220975394296854086800, 10.06040573325668452982898705079, 11.04369002459883656594845679056

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