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Properties

Label 2-13-13.8-c10-0-9
Degree $2$
Conductor $13$
Sign $-0.852 - 0.523i$
Analytic cond. $8.25964$
Root an. cond. $2.87395$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−22.7 − 22.7i)2-s + 85.0·3-s + 13.4i·4-s + (−1.40e3 − 1.40e3i)5-s + (−1.93e3 − 1.93e3i)6-s + (1.76e4 − 1.76e4i)7-s + (−2.30e4 + 2.30e4i)8-s − 5.18e4·9-s + 6.38e4i·10-s + (−1.56e5 + 1.56e5i)11-s + 1.14e3i·12-s + (−9.42e4 + 3.59e5i)13-s − 8.01e5·14-s + (−1.19e5 − 1.19e5i)15-s + 1.06e6·16-s − 1.06e6i·17-s + ⋯
L(s)  = 1  + (−0.711 − 0.711i)2-s + 0.350·3-s + 0.0131i·4-s + (−0.448 − 0.448i)5-s + (−0.249 − 0.249i)6-s + (1.04 − 1.04i)7-s + (−0.702 + 0.702i)8-s − 0.877·9-s + 0.638i·10-s + (−0.968 + 0.968i)11-s + 0.00458i·12-s + (−0.253 + 0.967i)13-s − 1.49·14-s + (−0.156 − 0.156i)15-s + 1.01·16-s − 0.749i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.523i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.852 - 0.523i$
Analytic conductor: \(8.25964\)
Root analytic conductor: \(2.87395\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :5),\ -0.852 - 0.523i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.126563 + 0.448024i\)
\(L(\frac12)\) \(\approx\) \(0.126563 + 0.448024i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (9.42e4 - 3.59e5i)T \)
good2 \( 1 + (22.7 + 22.7i)T + 1.02e3iT^{2} \)
3 \( 1 - 85.0T + 5.90e4T^{2} \)
5 \( 1 + (1.40e3 + 1.40e3i)T + 9.76e6iT^{2} \)
7 \( 1 + (-1.76e4 + 1.76e4i)T - 2.82e8iT^{2} \)
11 \( 1 + (1.56e5 - 1.56e5i)T - 2.59e10iT^{2} \)
17 \( 1 + 1.06e6iT - 2.01e12T^{2} \)
19 \( 1 + (2.58e6 + 2.58e6i)T + 6.13e12iT^{2} \)
23 \( 1 - 7.02e6iT - 4.14e13T^{2} \)
29 \( 1 - 1.57e7T + 4.20e14T^{2} \)
31 \( 1 + (2.05e7 + 2.05e7i)T + 8.19e14iT^{2} \)
37 \( 1 + (-3.47e7 + 3.47e7i)T - 4.80e15iT^{2} \)
41 \( 1 + (7.78e7 + 7.78e7i)T + 1.34e16iT^{2} \)
43 \( 1 - 6.47e7iT - 2.16e16T^{2} \)
47 \( 1 + (-2.50e8 + 2.50e8i)T - 5.25e16iT^{2} \)
53 \( 1 + 2.12e8T + 1.74e17T^{2} \)
59 \( 1 + (3.79e8 - 3.79e8i)T - 5.11e17iT^{2} \)
61 \( 1 + 7.59e8T + 7.13e17T^{2} \)
67 \( 1 + (6.35e8 + 6.35e8i)T + 1.82e18iT^{2} \)
71 \( 1 + (1.58e9 + 1.58e9i)T + 3.25e18iT^{2} \)
73 \( 1 + (-2.09e9 + 2.09e9i)T - 4.29e18iT^{2} \)
79 \( 1 - 3.56e9T + 9.46e18T^{2} \)
83 \( 1 + (3.11e9 + 3.11e9i)T + 1.55e19iT^{2} \)
89 \( 1 + (-5.13e7 + 5.13e7i)T - 3.11e19iT^{2} \)
97 \( 1 + (5.12e9 + 5.12e9i)T + 7.37e19iT^{2} \)
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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

−17.05658592605804062217455644759, −15.10704592929675214333150655693, −13.83874516942879088158598216858, −11.79077586254674406364616791222, −10.69919097740862250164840769617, −9.087402341576022235647184654659, −7.69319335966142455423584320050, −4.73737529793680518346707689634, −2.13736071835861634287856148815, −0.27118077570373820846155525729, 2.92836005058582918686561857458, 5.82318531419981215726165496205, 8.095196551121475509091306248627, 8.466502846337141903322502409773, 10.82950063444903323301326085194, 12.46167633223957314208326978346, 14.62395360271688749087280464285, 15.43348761820830241155969169910, 16.94331828188014475572066039734, 18.18565889468989424930646392600

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