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Properties

Label 2-26-13.12-c7-0-7
Degree $2$
Conductor $26$
Sign $0.696 + 0.717i$
Analytic cond. $8.12201$
Root an. cond. $2.84991$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8i·2-s + 37.3·3-s − 64·4-s − 477. i·5-s + 298. i·6-s − 855. i·7-s − 512i·8-s − 791.·9-s + 3.81e3·10-s + 1.40e3i·11-s − 2.39e3·12-s + (5.68e3 − 5.51e3i)13-s + 6.84e3·14-s − 1.78e4i·15-s + 4.09e3·16-s + 2.45e4·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.798·3-s − 0.5·4-s − 1.70i·5-s + 0.564i·6-s − 0.942i·7-s − 0.353i·8-s − 0.361·9-s + 1.20·10-s + 0.319i·11-s − 0.399·12-s + (0.717 − 0.696i)13-s + 0.666·14-s − 1.36i·15-s + 0.250·16-s + 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.696 + 0.717i$
Analytic conductor: \(8.12201\)
Root analytic conductor: \(2.84991\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :7/2),\ 0.696 + 0.717i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.70342 - 0.720395i\)
\(L(\frac12)\) \(\approx\) \(1.70342 - 0.720395i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 8iT \)
13 \( 1 + (-5.68e3 + 5.51e3i)T \)
good3 \( 1 - 37.3T + 2.18e3T^{2} \)
5 \( 1 + 477. iT - 7.81e4T^{2} \)
7 \( 1 + 855. iT - 8.23e5T^{2} \)
11 \( 1 - 1.40e3iT - 1.94e7T^{2} \)
17 \( 1 - 2.45e4T + 4.10e8T^{2} \)
19 \( 1 - 1.68e4iT - 8.93e8T^{2} \)
23 \( 1 + 5.41e4T + 3.40e9T^{2} \)
29 \( 1 - 1.80e5T + 1.72e10T^{2} \)
31 \( 1 - 7.30e4iT - 2.75e10T^{2} \)
37 \( 1 + 3.10e5iT - 9.49e10T^{2} \)
41 \( 1 - 4.05e5iT - 1.94e11T^{2} \)
43 \( 1 - 6.26e5T + 2.71e11T^{2} \)
47 \( 1 + 1.19e6iT - 5.06e11T^{2} \)
53 \( 1 + 1.10e6T + 1.17e12T^{2} \)
59 \( 1 - 2.21e6iT - 2.48e12T^{2} \)
61 \( 1 - 2.71e6T + 3.14e12T^{2} \)
67 \( 1 - 2.35e6iT - 6.06e12T^{2} \)
71 \( 1 - 2.11e6iT - 9.09e12T^{2} \)
73 \( 1 + 2.97e5iT - 1.10e13T^{2} \)
79 \( 1 - 6.13e6T + 1.92e13T^{2} \)
83 \( 1 - 5.86e6iT - 2.71e13T^{2} \)
89 \( 1 + 5.02e6iT - 4.42e13T^{2} \)
97 \( 1 + 4.56e6iT - 8.07e13T^{2} \)
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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

−15.92892410736979828736165091143, −14.36130520295957117566190647161, −13.45677977678945108519463731574, −12.31714091245178779600752055305, −9.977713306865509884469271761493, −8.598352537234478140196673415681, −7.84076489063281334773279257421, −5.56405511157667508443550690848, −3.94733861602606848941103630049, −0.931485827143625481202381971287, 2.40346642946976048599556956035, 3.38996153153598498202070648214, 6.15091331345553233560726328940, 8.089187157493919750689835335781, 9.490701775732310023675974372551, 10.90165701987149241149170707337, 11.94822014442688558657330938376, 13.89902401990644981531669282115, 14.42299995182693015976401961041, 15.66155340028181431469251293920

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