Properties

Degree 2
Conductor 13
Sign $0.999 + 0.0201i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.99 + 7.44i)2-s + (25.6 − 44.3i)3-s + (3.99 − 2.30i)4-s + (−23.8 + 23.8i)5-s + (381. + 102. i)6-s + (−75.6 + 282. i)7-s + (373. + 373. i)8-s + (−949. − 1.64e3i)9-s + (−224. − 129. i)10-s + (−427. + 114. i)11-s − 236. i·12-s + (40.6 + 2.19e3i)13-s − 2.25e3·14-s + (446. + 1.66e3i)15-s + (−1.88e3 + 3.27e3i)16-s + (2.79e3 − 1.61e3i)17-s + ⋯
L(s)  = 1  + (0.249 + 0.930i)2-s + (0.949 − 1.64i)3-s + (0.0624 − 0.0360i)4-s + (−0.190 + 0.190i)5-s + (1.76 + 0.473i)6-s + (−0.220 + 0.822i)7-s + (0.730 + 0.730i)8-s + (−1.30 − 2.25i)9-s + (−0.224 − 0.129i)10-s + (−0.321 + 0.0860i)11-s − 0.137i·12-s + (0.0184 + 0.999i)13-s − 0.820·14-s + (0.132 + 0.493i)15-s + (−0.461 + 0.799i)16-s + (0.569 − 0.328i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0201i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.999 + 0.0201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.999 + 0.0201i$
motivic weight  =  \(6\)
character  :  $\chi_{13} (6, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3),\ 0.999 + 0.0201i)$
$L(\frac{7}{2})$  $\approx$  $2.00927 - 0.0202244i$
$L(\frac12)$  $\approx$  $2.00927 - 0.0202244i$
$L(4)$   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 \( 1 + (-40.6 - 2.19e3i)T \)
good2 \( 1 + (-1.99 - 7.44i)T + (-55.4 + 32i)T^{2} \)
3 \( 1 + (-25.6 + 44.3i)T + (-364.5 - 631. i)T^{2} \)
5 \( 1 + (23.8 - 23.8i)T - 1.56e4iT^{2} \)
7 \( 1 + (75.6 - 282. i)T + (-1.01e5 - 5.88e4i)T^{2} \)
11 \( 1 + (427. - 114. i)T + (1.53e6 - 8.85e5i)T^{2} \)
17 \( 1 + (-2.79e3 + 1.61e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (1.01e4 + 2.71e3i)T + (4.07e7 + 2.35e7i)T^{2} \)
23 \( 1 + (-2.63e3 - 1.52e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-1.73e4 + 3.00e4i)T + (-2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (-5.70e3 + 5.70e3i)T - 8.87e8iT^{2} \)
37 \( 1 + (-3.12e3 + 836. i)T + (2.22e9 - 1.28e9i)T^{2} \)
41 \( 1 + (1.43e4 + 5.34e4i)T + (-4.11e9 + 2.37e9i)T^{2} \)
43 \( 1 + (-5.81e4 + 3.35e4i)T + (3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (7.05e4 + 7.05e4i)T + 1.07e10iT^{2} \)
53 \( 1 + 4.12e4T + 2.21e10T^{2} \)
59 \( 1 + (-1.17e4 + 4.38e4i)T + (-3.65e10 - 2.10e10i)T^{2} \)
61 \( 1 + (-4.19e4 - 7.26e4i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-4.13e4 - 1.54e5i)T + (-7.83e10 + 4.52e10i)T^{2} \)
71 \( 1 + (2.28e4 + 6.12e3i)T + (1.10e11 + 6.40e10i)T^{2} \)
73 \( 1 + (3.85e5 + 3.85e5i)T + 1.51e11iT^{2} \)
79 \( 1 + 5.12e5T + 2.43e11T^{2} \)
83 \( 1 + (-3.03e5 + 3.03e5i)T - 3.26e11iT^{2} \)
89 \( 1 + (9.61e4 - 2.57e4i)T + (4.30e11 - 2.48e11i)T^{2} \)
97 \( 1 + (-3.98e5 - 1.06e5i)T + (7.21e11 + 4.16e11i)T^{2} \)
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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

−18.83920839364871887028636353387, −17.27641806156220811040816841406, −15.40939581252311802257877709670, −14.42813688176130137188640488125, −13.25300025924665816698727930579, −11.80167248237429243246988159521, −8.723988075808841555828116130196, −7.38398737291880216680947444930, −6.23579039679411809966436412974, −2.24056711708226962061180638885, 3.07179460636630205802292128163, 4.36997068098182732149961871170, 8.201118444098053551367565986665, 10.14413401581256238642419249964, 10.73245268350621054206875456498, 12.88852268193243841397238794385, 14.41524536796254631530769450118, 15.80388063685907925135717249437, 16.74629342102950177648914323582, 19.45795931174931931957093350874

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