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Properties

Label 2-13-13.8-c14-0-15
Degree $2$
Conductor $13$
Sign $-0.940 - 0.338i$
Analytic cond. $16.1627$
Root an. cond. $4.02029$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−20.7 − 20.7i)2-s + 2.10e3·3-s − 1.55e4i·4-s + (−6.37e4 − 6.37e4i)5-s + (−4.36e4 − 4.36e4i)6-s + (−5.24e5 + 5.24e5i)7-s + (−6.61e5 + 6.61e5i)8-s − 3.55e5·9-s + 2.64e6i·10-s + (1.16e7 − 1.16e7i)11-s − 3.26e7i·12-s + (−3.21e6 + 6.26e7i)13-s + 2.17e7·14-s + (−1.34e8 − 1.34e8i)15-s − 2.26e8·16-s + 7.70e8i·17-s + ⋯
L(s)  = 1  + (−0.162 − 0.162i)2-s + 0.962·3-s − 0.947i·4-s + (−0.815 − 0.815i)5-s + (−0.155 − 0.155i)6-s + (−0.637 + 0.637i)7-s + (−0.315 + 0.315i)8-s − 0.0743·9-s + 0.264i·10-s + (0.599 − 0.599i)11-s − 0.911i·12-s + (−0.0512 + 0.998i)13-s + 0.206·14-s + (−0.785 − 0.785i)15-s − 0.845·16-s + 1.87i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.338i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.940 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.940 - 0.338i$
Analytic conductor: \(16.1627\)
Root analytic conductor: \(4.02029\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :7),\ -0.940 - 0.338i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.0852985 + 0.489149i\)
\(L(\frac12)\) \(\approx\) \(0.0852985 + 0.489149i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.21e6 - 6.26e7i)T \)
good2 \( 1 + (20.7 + 20.7i)T + 1.63e4iT^{2} \)
3 \( 1 - 2.10e3T + 4.78e6T^{2} \)
5 \( 1 + (6.37e4 + 6.37e4i)T + 6.10e9iT^{2} \)
7 \( 1 + (5.24e5 - 5.24e5i)T - 6.78e11iT^{2} \)
11 \( 1 + (-1.16e7 + 1.16e7i)T - 3.79e14iT^{2} \)
17 \( 1 - 7.70e8iT - 1.68e17T^{2} \)
19 \( 1 + (8.30e8 + 8.30e8i)T + 7.99e17iT^{2} \)
23 \( 1 + 2.80e9iT - 1.15e19T^{2} \)
29 \( 1 + 1.62e10T + 2.97e20T^{2} \)
31 \( 1 + (1.57e10 + 1.57e10i)T + 7.56e20iT^{2} \)
37 \( 1 + (-9.20e10 + 9.20e10i)T - 9.01e21iT^{2} \)
41 \( 1 + (5.99e10 + 5.99e10i)T + 3.79e22iT^{2} \)
43 \( 1 + 6.67e10iT - 7.38e22T^{2} \)
47 \( 1 + (-2.70e11 + 2.70e11i)T - 2.56e23iT^{2} \)
53 \( 1 + 2.02e12T + 1.37e24T^{2} \)
59 \( 1 + (-7.69e11 + 7.69e11i)T - 6.19e24iT^{2} \)
61 \( 1 - 3.04e12T + 9.87e24T^{2} \)
67 \( 1 + (-5.59e11 - 5.59e11i)T + 3.67e25iT^{2} \)
71 \( 1 + (1.33e12 + 1.33e12i)T + 8.27e25iT^{2} \)
73 \( 1 + (-1.23e13 + 1.23e13i)T - 1.22e26iT^{2} \)
79 \( 1 + 1.94e13T + 3.68e26T^{2} \)
83 \( 1 + (1.55e13 + 1.55e13i)T + 7.36e26iT^{2} \)
89 \( 1 + (2.56e13 - 2.56e13i)T - 1.95e27iT^{2} \)
97 \( 1 + (-5.32e13 - 5.32e13i)T + 6.52e27iT^{2} \)
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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

−15.35568807125759674021637558267, −14.43479136965324002913464101533, −12.80701800379969346439511111038, −11.20778194219272280463990070669, −9.199264673204045774000460368014, −8.552104052396308275400538951690, −6.14230727680387722651089641524, −4.03995042667780009096284292357, −2.05700470429290466783398710750, −0.17156284476192240522516117606, 2.90851622875753549347429356452, 3.75596410374208872735399954931, 7.05906790071252178431384643037, 7.930581258779861984937156603681, 9.542456452661734871968627048279, 11.49256352330072119833003943701, 13.03296280957889825593380157587, 14.46294361040034789063480308796, 15.66593018346464359457568354316, 17.04337633126514804440554802259

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