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Properties

Label 2-26e2-676.451-c0-0-0
Degree $2$
Conductor $676$
Sign $-0.573 - 0.819i$
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  + (0.799 + 0.600i)2-s + (0.278 + 0.960i)4-s + (−1.62 + 0.855i)5-s + (−0.354 + 0.935i)8-s + (−0.919 + 0.391i)9-s + (−1.81 − 0.295i)10-s + (0.987 + 0.160i)13-s + (−0.845 + 0.534i)16-s + (0.549 − 0.0892i)17-s + (−0.970 − 0.239i)18-s + (−1.27 − 1.32i)20-s + (1.35 − 1.96i)25-s + (0.692 + 0.721i)26-s + (1.51 + 1.13i)29-s + (−0.996 − 0.0804i)32-s + ⋯
L(s)  = 1  + (0.799 + 0.600i)2-s + (0.278 + 0.960i)4-s + (−1.62 + 0.855i)5-s + (−0.354 + 0.935i)8-s + (−0.919 + 0.391i)9-s + (−1.81 − 0.295i)10-s + (0.987 + 0.160i)13-s + (−0.845 + 0.534i)16-s + (0.549 − 0.0892i)17-s + (−0.970 − 0.239i)18-s + (−1.27 − 1.32i)20-s + (1.35 − 1.96i)25-s + (0.692 + 0.721i)26-s + (1.51 + 1.13i)29-s + (−0.996 − 0.0804i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.573 - 0.819i$
Analytic conductor: \(0.337367\)
Root analytic conductor: \(0.580833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :0),\ -0.573 - 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.030966846\)
\(L(\frac12)\) \(\approx\) \(1.030966846\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.799 - 0.600i)T \)
13 \( 1 + (-0.987 - 0.160i)T \)
good3 \( 1 + (0.919 - 0.391i)T^{2} \)
5 \( 1 + (1.62 - 0.855i)T + (0.568 - 0.822i)T^{2} \)
7 \( 1 + (-0.948 + 0.316i)T^{2} \)
11 \( 1 + (-0.692 - 0.721i)T^{2} \)
17 \( 1 + (-0.549 + 0.0892i)T + (0.948 - 0.316i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1.51 - 1.13i)T + (0.278 + 0.960i)T^{2} \)
31 \( 1 + (0.354 - 0.935i)T^{2} \)
37 \( 1 + (-0.0802 + 0.00648i)T + (0.987 - 0.160i)T^{2} \)
41 \( 1 + (0.319 + 1.56i)T + (-0.919 + 0.391i)T^{2} \)
43 \( 1 + (-0.987 - 0.160i)T^{2} \)
47 \( 1 + (-0.885 + 0.464i)T^{2} \)
53 \( 1 + (0.700 - 1.84i)T + (-0.748 - 0.663i)T^{2} \)
59 \( 1 + (-0.428 - 0.903i)T^{2} \)
61 \( 1 + (-0.253 + 0.309i)T + (-0.200 - 0.979i)T^{2} \)
67 \( 1 + (0.845 + 0.534i)T^{2} \)
71 \( 1 + (-0.799 - 0.600i)T^{2} \)
73 \( 1 + (0.120 + 0.992i)T + (-0.970 + 0.239i)T^{2} \)
79 \( 1 + (-0.885 + 0.464i)T^{2} \)
83 \( 1 + (-0.120 - 0.992i)T^{2} \)
89 \( 1 + (0.885 - 1.53i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.0457 + 1.13i)T + (-0.996 + 0.0804i)T^{2} \)
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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

−11.06211580401289579998807118626, −10.67324587770235558995438206874, −8.758859587331672202150334307999, −8.231033114602887516407831288433, −7.40403515374296025468821264350, −6.66331521106657690430492529302, −5.64968332564107123466972341969, −4.48482872051563442467399278504, −3.53804136343004117371272855499, −2.84486073662526223653665214946, 0.950572021859315465165074134956, 3.00595860172875139765553262075, 3.82247358949072966105623407241, 4.65720903195507977096335184956, 5.67537449772706237535023185281, 6.67420190445387652718753657308, 8.036392054857826057155091872100, 8.538599907836135298573857362825, 9.622940617471127929578656920185, 10.77414090595126364727355631079

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