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Properties

Label 2-416-13.6-c0-0-0
Degree $2$
Conductor $416$
Sign $0.295 - 0.955i$
Analytic cond. $0.207611$
Root an. cond. $0.455643$
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  + (−1.36 + 1.36i)5-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)17-s − 2.73i·25-s + (0.866 − 1.5i)29-s + (−0.5 + 0.133i)37-s + (−0.133 − 0.5i)41-s + (−1.86 − 0.499i)45-s + (0.866 + 0.5i)49-s + 53-s + (−0.5 − 0.866i)61-s + (−1.86 + 0.499i)65-s + (0.366 + 0.366i)73-s + (−0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)5-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)17-s − 2.73i·25-s + (0.866 − 1.5i)29-s + (−0.5 + 0.133i)37-s + (−0.133 − 0.5i)41-s + (−1.86 − 0.499i)45-s + (0.866 + 0.5i)49-s + 53-s + (−0.5 − 0.866i)61-s + (−1.86 + 0.499i)65-s + (0.366 + 0.366i)73-s + (−0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.295 - 0.955i$
Analytic conductor: \(0.207611\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :0),\ 0.295 - 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6893850888\)
\(L(\frac12)\) \(\approx\) \(0.6893850888\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-0.866 - 0.5i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - T + T^{2} \)
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 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
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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

−11.41624194969527144345767400198, −10.82371162630220627877010433775, −10.15474665355562215693007704732, −8.633953890918256721347139708589, −7.85577955530207773043555271028, −7.02367063293957945467988403344, −6.22649325618763554957574536774, −4.47821608295843007777231169554, −3.73100072127709119730062882403, −2.37386726625514175552389770819, 1.04679569147095475428073483002, 3.43994054432410308375508938141, 4.30557116395216178513903876179, 5.25206990133273080396111963400, 6.68622880738325413153013292019, 7.65336820753368910381754746136, 8.719256151719974237311947490420, 9.019446987177833647461266134652, 10.43102001272594528341710527739, 11.47238592024115589298265815116

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