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Properties

Label 2-476-476.135-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.7966479076\)
\(L(\frac12)\) \(\approx\) \(0.7966479076\)
\(L(1)\) 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

−11.09535139220431389658158850983, −10.26181528170142932020872155541, −9.189168278053678897071393153337, −7.79619662493473746985943868253, −6.81473999656473342397938340318, −6.28913316104520850601139058727, −4.97159460935988884613838754383, −3.97962228810159768289381329074, −2.17144802826572071518473476174, −1.08987278157440062378599310698, 3.39453644134763664225732626843, 4.01465038350638189169196351049, 5.27576068283942987298436104519, 5.91439906930685071175477783227, 6.47252090941402724136107794406, 8.395387336984686484425394313451, 8.855694864116422952925778850403, 9.699092736512341396188385167651, 11.13484282716842273688163895251, 11.36622786111648245464148500183

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