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Properties

Label 2-476-476.135-c0-0-0
Degree $2$
Conductor $476$
Sign $-0.0633 - 0.997i$
Analytic cond. $0.237554$
Root an. cond. $0.487396$
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  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s + 7-s + 0.999·8-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s − 13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)21-s − 0.999·22-s + (−1 + 1.73i)23-s + (0.499 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s + 7-s + 0.999·8-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s − 13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)21-s − 0.999·22-s + (−1 + 1.73i)23-s + (0.499 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $-0.0633 - 0.997i$
Analytic conductor: \(0.237554\)
Root analytic conductor: \(0.487396\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :0),\ -0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8221604333\)
\(L(\frac12)\) \(\approx\) \(0.8221604333\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 - T \)
17 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.28398873705967760862828976770, −10.05548438281783237966502687101, −9.607228205255160941572669737650, −8.903051136991857499508378943688, −7.74353239022407903109732748954, −7.22002136838148515468643957817, −5.80831173930127471811794986542, −4.71149935488822195409814089193, −4.09113388242388090002124482833, −2.02529007585297983418415934769, 1.54988162817002127811061103176, 2.46078269995862118880920473877, 3.90385438107837015642905449248, 5.06584261902738161210797675358, 6.70844418690888930519692428745, 7.68277022536209649363553102073, 8.380645916098363946946391612502, 8.964896026681887126191137317872, 10.31372492094595677668848111187, 10.93117195442424377632432964820

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