Properties

Degree 2
Conductor 13
Sign $-0.911 - 0.411i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.92 + 10.9i)2-s + (14.1 + 24.4i)3-s + (−55.2 − 31.9i)4-s + (−47.2 − 47.2i)5-s + (−308. + 82.5i)6-s + (27.7 + 103. i)7-s + (−1.56 + 1.56i)8-s + (−33.5 + 58.1i)9-s + (654. − 377. i)10-s + (2.24e3 + 601. i)11-s − 1.80e3i·12-s + (−980. + 1.96e3i)13-s − 1.21e3·14-s + (488. − 1.82e3i)15-s + (−2.05e3 − 3.55e3i)16-s + (2.97e3 + 1.71e3i)17-s + ⋯
L(s)  = 1  + (−0.365 + 1.36i)2-s + (0.522 + 0.905i)3-s + (−0.863 − 0.498i)4-s + (−0.378 − 0.378i)5-s + (−1.42 + 0.382i)6-s + (0.0809 + 0.302i)7-s + (−0.00306 + 0.00306i)8-s + (−0.0460 + 0.0797i)9-s + (0.654 − 0.377i)10-s + (1.68 + 0.452i)11-s − 1.04i·12-s + (−0.446 + 0.894i)13-s − 0.441·14-s + (0.144 − 0.539i)15-s + (−0.501 − 0.868i)16-s + (0.605 + 0.349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $-0.911 - 0.411i$
motivic weight  =  \(6\)
character  :  $\chi_{13} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3),\ -0.911 - 0.411i)$
$L(\frac{7}{2})$  $\approx$  $0.266770 + 1.23994i$
$L(\frac12)$  $\approx$  $0.266770 + 1.23994i$
$L(4)$   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 \( 1 + (980. - 1.96e3i)T \)
good2 \( 1 + (2.92 - 10.9i)T + (-55.4 - 32i)T^{2} \)
3 \( 1 + (-14.1 - 24.4i)T + (-364.5 + 631. i)T^{2} \)
5 \( 1 + (47.2 + 47.2i)T + 1.56e4iT^{2} \)
7 \( 1 + (-27.7 - 103. i)T + (-1.01e5 + 5.88e4i)T^{2} \)
11 \( 1 + (-2.24e3 - 601. i)T + (1.53e6 + 8.85e5i)T^{2} \)
17 \( 1 + (-2.97e3 - 1.71e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (4.73e3 - 1.26e3i)T + (4.07e7 - 2.35e7i)T^{2} \)
23 \( 1 + (-1.68e4 + 9.70e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (1.40e4 + 2.43e4i)T + (-2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (954. + 954. i)T + 8.87e8iT^{2} \)
37 \( 1 + (5.81e3 + 1.55e3i)T + (2.22e9 + 1.28e9i)T^{2} \)
41 \( 1 + (1.60e4 - 5.98e4i)T + (-4.11e9 - 2.37e9i)T^{2} \)
43 \( 1 + (4.57e4 + 2.64e4i)T + (3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (2.83e4 - 2.83e4i)T - 1.07e10iT^{2} \)
53 \( 1 - 4.03e4T + 2.21e10T^{2} \)
59 \( 1 + (9.69e4 + 3.61e5i)T + (-3.65e10 + 2.10e10i)T^{2} \)
61 \( 1 + (4.69e4 - 8.13e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (9.25e4 - 3.45e5i)T + (-7.83e10 - 4.52e10i)T^{2} \)
71 \( 1 + (2.88e5 - 7.71e4i)T + (1.10e11 - 6.40e10i)T^{2} \)
73 \( 1 + (-1.97e5 + 1.97e5i)T - 1.51e11iT^{2} \)
79 \( 1 + 7.57e5T + 2.43e11T^{2} \)
83 \( 1 + (-1.21e5 - 1.21e5i)T + 3.26e11iT^{2} \)
89 \( 1 + (-1.33e6 - 3.58e5i)T + (4.30e11 + 2.48e11i)T^{2} \)
97 \( 1 + (2.42e5 - 6.48e4i)T + (7.21e11 - 4.16e11i)T^{2} \)
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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

−19.01584430626729716214699230949, −17.16968480670210015832028124220, −16.42072938474403799239867744519, −15.01605743622932183969808591874, −14.54401683903381597010433462566, −11.99160421754823025457173669245, −9.510364341660747234572042670652, −8.574658859502137880917347718480, −6.66991113238713904922777921847, −4.38430297715524655434866293584, 1.26103799657995284059154825787, 3.27650908085141146546162415786, 7.17092819742790989896186751772, 9.019364436721934136570258266822, 10.79421565293172245760998582998, 12.05666218488786415962343430744, 13.30767801430774056565815084534, 14.82634590476207245666303205327, 17.15842480316884647315911466004, 18.65793487480210094301534554075

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