Properties

Degree 2
Conductor 13
Sign $0.704 + 0.709i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.28 − 3.95i)2-s + (−4.34 + 7.52i)3-s + (−6.40 − 11.0i)4-s + 2.80·5-s + (19.8 + 34.3i)6-s + (−4.78 − 8.28i)7-s − 21.9·8-s + (−24.2 − 41.9i)9-s + (6.40 − 11.0i)10-s + (−19.7 + 34.1i)11-s + 111.·12-s + (40.5 − 23.5i)13-s − 43.6·14-s + (−12.1 + 21.1i)15-s + (1.21 − 2.09i)16-s + (−1.00 − 1.74i)17-s + ⋯
L(s)  = 1  + (0.806 − 1.39i)2-s + (−0.835 + 1.44i)3-s + (−0.800 − 1.38i)4-s + 0.251·5-s + (1.34 + 2.33i)6-s + (−0.258 − 0.447i)7-s − 0.969·8-s + (−0.896 − 1.55i)9-s + (0.202 − 0.350i)10-s + (−0.540 + 0.935i)11-s + 2.67·12-s + (0.864 − 0.502i)13-s − 0.832·14-s + (−0.209 + 0.363i)15-s + (0.0189 − 0.0327i)16-s + (−0.0143 − 0.0248i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.704 + 0.709i$
motivic weight  =  \(3\)
character  :  $\chi_{13} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3/2),\ 0.704 + 0.709i)$
$L(2)$  $\approx$  $0.999975 - 0.416327i$
$L(\frac12)$  $\approx$  $0.999975 - 0.416327i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 13$, \(F_p\) is a polynomial of degree 2. If $p = 13$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad13 \( 1 + (-40.5 + 23.5i)T \)
good2 \( 1 + (-2.28 + 3.95i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (4.34 - 7.52i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 2.80T + 125T^{2} \)
7 \( 1 + (4.78 + 8.28i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (19.7 - 34.1i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (1.00 + 1.74i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-30.0 - 52.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (2.23 - 3.87i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (70.3 - 121. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 136.T + 2.97e4T^{2} \)
37 \( 1 + (-92.8 + 160. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (155. - 268. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (213. + 370. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 258.T + 1.03e5T^{2} \)
53 \( 1 - 612.T + 1.48e5T^{2} \)
59 \( 1 + (-258. - 448. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-80.6 - 139. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-24.9 + 43.2i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (139. + 242. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 467.T + 3.89e5T^{2} \)
79 \( 1 - 37.5T + 4.93e5T^{2} \)
83 \( 1 + 76.1T + 5.71e5T^{2} \)
89 \( 1 + (101. - 175. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-587. - 1.01e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−20.06374713047679510172826553572, −18.06018248216009198275540077030, −16.47930155832714051475863650637, −15.09259625175710776437642716566, −13.37139333206078506665771272037, −11.85655960895952525637217088232, −10.56005695066024981131704880800, −9.876914585513506265401796361910, −5.36503335546656995246130403068, −3.83978155581531791135561886183, 5.64843544798853391137423986062, 6.58529346849352433488911864635, 8.137050051906496283020045415523, 11.56521870450852786934786447720, 13.14982582062895793753971993219, 13.74681360234623211552995837401, 15.69412939009807016563799113631, 16.84117275849509255145122153563, 18.00142307987297076143490870653, 19.07593813479511627054658402635

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