Properties

Degree 2
Conductor 13
Sign $-0.916 - 0.400i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.77 − 1.27i)2-s + (4.54 + 7.86i)3-s + (−34.2 − 19.7i)4-s + (−109. + 109. i)5-s + (−11.6 − 43.3i)6-s + (−504. + 135. i)7-s + (361. + 361. i)8-s + (323. − 559. i)9-s + (663. − 383. i)10-s + (246. − 919. i)11-s − 359. i·12-s + (300. + 2.17e3i)13-s + 2.58e3·14-s + (−1.36e3 − 364. i)15-s + (1.54 + 2.68i)16-s + (−5.34e3 − 3.08e3i)17-s + ⋯
L(s)  = 1  + (−0.596 − 0.159i)2-s + (0.168 + 0.291i)3-s + (−0.535 − 0.309i)4-s + (−0.877 + 0.877i)5-s + (−0.0538 − 0.200i)6-s + (−1.47 + 0.394i)7-s + (0.706 + 0.706i)8-s + (0.443 − 0.767i)9-s + (0.663 − 0.383i)10-s + (0.185 − 0.691i)11-s − 0.208i·12-s + (0.136 + 0.990i)13-s + 0.940·14-s + (−0.403 − 0.108i)15-s + (0.000378 + 0.000655i)16-s + (−1.08 − 0.628i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $-0.916 - 0.400i$
motivic weight  =  \(6\)
character  :  $\chi_{13} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3),\ -0.916 - 0.400i)$
$L(\frac{7}{2})$  $\approx$  $0.0449915 + 0.215211i$
$L(\frac12)$  $\approx$  $0.0449915 + 0.215211i$
$L(4)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 13$,\(F_p(T)\) is a polynomial of degree 2. If $p = 13$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 + (-300. - 2.17e3i)T \)
good2 \( 1 + (4.77 + 1.27i)T + (55.4 + 32i)T^{2} \)
3 \( 1 + (-4.54 - 7.86i)T + (-364.5 + 631. i)T^{2} \)
5 \( 1 + (109. - 109. i)T - 1.56e4iT^{2} \)
7 \( 1 + (504. - 135. i)T + (1.01e5 - 5.88e4i)T^{2} \)
11 \( 1 + (-246. + 919. i)T + (-1.53e6 - 8.85e5i)T^{2} \)
17 \( 1 + (5.34e3 + 3.08e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-1.06e3 - 3.95e3i)T + (-4.07e7 + 2.35e7i)T^{2} \)
23 \( 1 + (9.99e3 - 5.76e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (186. + 322. i)T + (-2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (1.08e4 - 1.08e4i)T - 8.87e8iT^{2} \)
37 \( 1 + (2.23e4 - 8.34e4i)T + (-2.22e9 - 1.28e9i)T^{2} \)
41 \( 1 + (-1.61e3 - 432. i)T + (4.11e9 + 2.37e9i)T^{2} \)
43 \( 1 + (8.83e4 + 5.10e4i)T + (3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (2.76e4 + 2.76e4i)T + 1.07e10iT^{2} \)
53 \( 1 - 2.39e5T + 2.21e10T^{2} \)
59 \( 1 + (-1.20e5 + 3.22e4i)T + (3.65e10 - 2.10e10i)T^{2} \)
61 \( 1 + (-3.82e4 + 6.61e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (3.54e5 + 9.50e4i)T + (7.83e10 + 4.52e10i)T^{2} \)
71 \( 1 + (1.11e5 + 4.17e5i)T + (-1.10e11 + 6.40e10i)T^{2} \)
73 \( 1 + (-3.43e5 - 3.43e5i)T + 1.51e11iT^{2} \)
79 \( 1 + 5.59e5T + 2.43e11T^{2} \)
83 \( 1 + (5.51e4 - 5.51e4i)T - 3.26e11iT^{2} \)
89 \( 1 + (1.89e5 - 7.08e5i)T + (-4.30e11 - 2.48e11i)T^{2} \)
97 \( 1 + (-3.28e5 - 1.22e6i)T + (-7.21e11 + 4.16e11i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.99002630745060446217596343305, −18.32072516093902756586779196758, −16.32073279641383157136361579313, −15.20183171961515563360711867539, −13.66983811116506301684615556211, −11.70398011910301577808959017700, −10.03988782084036896156845757175, −8.904646485132924383134576642328, −6.67356704279308506767378270564, −3.66209977729891911120260313639, 0.20234623542748597957589959277, 4.18693168959237015533867176305, 7.29313888701180730309696394275, 8.631495046570037573677865082408, 10.15256339774363322707231863426, 12.67900780824347180402812752001, 13.21102654030892646228921867011, 15.75513084031562921180013702884, 16.56107991459155638963349349351, 17.97698837918069913144722514796

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