Properties

Degree 2
Conductor 13
Sign $0.548 - 0.836i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.89 + 1.57i)2-s + (20.5 + 35.6i)3-s + (−23.2 − 13.4i)4-s + (−37.1 + 37.1i)5-s + (64.9 + 242. i)6-s + (588. − 157. i)7-s + (−391. − 391. i)8-s + (−481. + 834. i)9-s + (−277. + 160. i)10-s + (357. − 1.33e3i)11-s − 1.10e3i·12-s + (−2.19e3 − 67.7i)13-s + 3.71e3·14-s + (−2.09e3 − 560. i)15-s + (−830. − 1.43e3i)16-s + (542. + 313. i)17-s + ⋯
L(s)  = 1  + (0.736 + 0.197i)2-s + (0.761 + 1.31i)3-s + (−0.362 − 0.209i)4-s + (−0.297 + 0.297i)5-s + (0.300 + 1.12i)6-s + (1.71 − 0.459i)7-s + (−0.764 − 0.764i)8-s + (−0.660 + 1.14i)9-s + (−0.277 + 0.160i)10-s + (0.268 − 1.00i)11-s − 0.638i·12-s + (−0.999 − 0.0308i)13-s + 1.35·14-s + (−0.619 − 0.165i)15-s + (−0.202 − 0.351i)16-s + (0.110 + 0.0637i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.548 - 0.836i$
motivic weight  =  \(6\)
character  :  $\chi_{13} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3),\ 0.548 - 0.836i)$
$L(\frac{7}{2})$  $\approx$  $1.89275 + 1.02246i$
$L(\frac12)$  $\approx$  $1.89275 + 1.02246i$
$L(4)$   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 \( 1 + (2.19e3 + 67.7i)T \)
good2 \( 1 + (-5.89 - 1.57i)T + (55.4 + 32i)T^{2} \)
3 \( 1 + (-20.5 - 35.6i)T + (-364.5 + 631. i)T^{2} \)
5 \( 1 + (37.1 - 37.1i)T - 1.56e4iT^{2} \)
7 \( 1 + (-588. + 157. i)T + (1.01e5 - 5.88e4i)T^{2} \)
11 \( 1 + (-357. + 1.33e3i)T + (-1.53e6 - 8.85e5i)T^{2} \)
17 \( 1 + (-542. - 313. i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-493. - 1.84e3i)T + (-4.07e7 + 2.35e7i)T^{2} \)
23 \( 1 + (8.02e3 - 4.63e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (1.51e4 + 2.63e4i)T + (-2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (-1.07e4 + 1.07e4i)T - 8.87e8iT^{2} \)
37 \( 1 + (1.50e4 - 5.62e4i)T + (-2.22e9 - 1.28e9i)T^{2} \)
41 \( 1 + (5.13e4 + 1.37e4i)T + (4.11e9 + 2.37e9i)T^{2} \)
43 \( 1 + (-2.24e4 - 1.29e4i)T + (3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (2.15e4 + 2.15e4i)T + 1.07e10iT^{2} \)
53 \( 1 + 2.19e5T + 2.21e10T^{2} \)
59 \( 1 + (-1.56e5 + 4.19e4i)T + (3.65e10 - 2.10e10i)T^{2} \)
61 \( 1 + (-2.01e5 + 3.48e5i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.49e4 + 3.99e3i)T + (7.83e10 + 4.52e10i)T^{2} \)
71 \( 1 + (-1.36e5 - 5.09e5i)T + (-1.10e11 + 6.40e10i)T^{2} \)
73 \( 1 + (-1.21e5 - 1.21e5i)T + 1.51e11iT^{2} \)
79 \( 1 - 3.26e5T + 2.43e11T^{2} \)
83 \( 1 + (3.42e5 - 3.42e5i)T - 3.26e11iT^{2} \)
89 \( 1 + (5.63e4 - 2.10e5i)T + (-4.30e11 - 2.48e11i)T^{2} \)
97 \( 1 + (-2.81e5 - 1.05e6i)T + (-7.21e11 + 4.16e11i)T^{2} \)
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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

−18.97424020230946054363467786281, −17.18876371469873033401227026474, −15.42788963385115945149304578131, −14.58651184819583043301134971040, −13.88831559750682118701133465385, −11.38972183564343554012350499190, −9.823472133038607007380979599700, −8.189054451887752547527675796261, −5.05670847824028678965765775136, −3.79370955995251489686209277756, 2.10514707109587005814953167834, 4.79037846808960750799694372832, 7.58908933605578771571178637324, 8.694694905354778705848244080309, 11.91584173104973382108759952022, 12.58846343251021896269934454372, 14.17257389902522449208563163630, 14.74823375790331240862478316627, 17.57697249120855917590488363625, 18.22694759422366141605353017843

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