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Properties

Label 2-26e2-676.35-c0-0-0
Degree $2$
Conductor $676$
Sign $0.132 + 0.991i$
Analytic cond. $0.337367$
Root an. cond. $0.580833$
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  + (0.692 − 0.721i)2-s + (−0.0402 − 0.999i)4-s + (0.316 − 0.457i)5-s + (−0.748 − 0.663i)8-s + (0.278 + 0.960i)9-s + (−0.111 − 0.545i)10-s + (−0.200 − 0.979i)13-s + (−0.996 + 0.0804i)16-s + (0.0161 − 0.0789i)17-s + (0.885 + 0.464i)18-s + (−0.470 − 0.297i)20-s + (0.244 + 0.645i)25-s + (−0.845 − 0.534i)26-s + (−1.27 + 1.32i)29-s + (−0.632 + 0.774i)32-s + ⋯
L(s)  = 1  + (0.692 − 0.721i)2-s + (−0.0402 − 0.999i)4-s + (0.316 − 0.457i)5-s + (−0.748 − 0.663i)8-s + (0.278 + 0.960i)9-s + (−0.111 − 0.545i)10-s + (−0.200 − 0.979i)13-s + (−0.996 + 0.0804i)16-s + (0.0161 − 0.0789i)17-s + (0.885 + 0.464i)18-s + (−0.470 − 0.297i)20-s + (0.244 + 0.645i)25-s + (−0.845 − 0.534i)26-s + (−1.27 + 1.32i)29-s + (−0.632 + 0.774i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(0.337367\)
Root analytic conductor: \(0.580833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :0),\ 0.132 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.344122613\)
\(L(\frac12)\) \(\approx\) \(1.344122613\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.692 + 0.721i)T \)
13 \( 1 + (0.200 + 0.979i)T \)
good3 \( 1 + (-0.278 - 0.960i)T^{2} \)
5 \( 1 + (-0.316 + 0.457i)T + (-0.354 - 0.935i)T^{2} \)
7 \( 1 + (0.919 + 0.391i)T^{2} \)
11 \( 1 + (0.845 + 0.534i)T^{2} \)
17 \( 1 + (-0.0161 + 0.0789i)T + (-0.919 - 0.391i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.27 - 1.32i)T + (-0.0402 - 0.999i)T^{2} \)
31 \( 1 + (0.748 + 0.663i)T^{2} \)
37 \( 1 + (0.542 + 0.664i)T + (-0.200 + 0.979i)T^{2} \)
41 \( 1 + (-1.10 - 0.832i)T + (0.278 + 0.960i)T^{2} \)
43 \( 1 + (0.200 + 0.979i)T^{2} \)
47 \( 1 + (-0.568 + 0.822i)T^{2} \)
53 \( 1 + (-0.299 - 0.265i)T + (0.120 + 0.992i)T^{2} \)
59 \( 1 + (-0.987 - 0.160i)T^{2} \)
61 \( 1 + (-1.51 - 0.506i)T + (0.799 + 0.600i)T^{2} \)
67 \( 1 + (0.996 + 0.0804i)T^{2} \)
71 \( 1 + (-0.692 + 0.721i)T^{2} \)
73 \( 1 + (-0.970 + 0.239i)T + (0.885 - 0.464i)T^{2} \)
79 \( 1 + (-0.568 + 0.822i)T^{2} \)
83 \( 1 + (0.970 - 0.239i)T^{2} \)
89 \( 1 + (0.568 - 0.983i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.304 - 0.640i)T + (-0.632 - 0.774i)T^{2} \)
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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

−10.71142409674171924120500085872, −9.822961754593031007359597160370, −9.050825410797154154780339932934, −7.905164687531826022481308989724, −6.87238880265330770131838199457, −5.46686959198033708824047982945, −5.18854899169540605095111830741, −3.94238613029829272087898040419, −2.74133377157108365266817907660, −1.52145358394408843262235377360, 2.30325776375068960906301405570, 3.61651755092903445131816218414, 4.42364729033299969006722317407, 5.67583233933788566735085428935, 6.50421011901631042146774806423, 7.07900652294644743889374241814, 8.136240579844616979407322856978, 9.164520913783472253519025985879, 9.835913901752934778615050118334, 11.12903250028556812405422461632

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