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Properties

Label 2-476-476.135-c0-0-1
Degree $2$
Conductor $476$
Sign $0.997 - 0.0633i$
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 + (0.866 + 1.5i)3-s + (−0.499 − 0.866i)4-s + 1.73·6-s + i·7-s − 0.999·8-s + (−1 + 1.73i)9-s + (−0.866 − 1.5i)11-s + (0.866 − 1.49i)12-s + 13-s + (0.866 + 0.5i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (1 + 1.73i)18-s + (−1.5 + 0.866i)21-s − 1.73·22-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.866 + 1.5i)3-s + (−0.499 − 0.866i)4-s + 1.73·6-s + i·7-s − 0.999·8-s + (−1 + 1.73i)9-s + (−0.866 − 1.5i)11-s + (0.866 − 1.49i)12-s + 13-s + (0.866 + 0.5i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (1 + 1.73i)18-s + (−1.5 + 0.866i)21-s − 1.73·22-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.997 - 0.0633i$
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: \((2,\ 476,\ (\ :0),\ 0.997 - 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.270683077\)
\(L(\frac12)\) \(\approx\) \(1.270683077\)
\(L(1)\) 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 \( 1 - iT \)
17 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.73T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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

−11.07003735318917306378206549789, −10.45471006190013588299365981310, −9.508112007167664038179505333331, −8.762677806028502181079938116642, −8.295134243651476894657254720512, −6.00991797545547029172835080837, −5.29448475235018583592176983778, −4.24242304153068494756273286680, −3.19847471020486908205697602242, −2.54572849399032149803958116538, 1.85648996883229754103561318628, 3.32401147816787240906483124784, 4.43083053375474411949538090420, 5.92906387769723280787642798314, 6.91102574826430035000289349049, 7.43124846454260907082139745504, 8.111145523124783815237780202687, 8.959114864365666946139483014187, 10.19184071162204878715903048090, 11.54774420531487042648510162461

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