Properties

Degree 2
Conductor 13
Sign $0.984 + 0.176i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.581 − 0.581i)2-s − 4.16·3-s + 3.32i·4-s + (3.58 − 3.58i)5-s + (−2.41 + 2.41i)6-s + (−4.58 − 4.58i)7-s + (4.25 + 4.25i)8-s + 8.32·9-s − 4.16i·10-s + (5.32 + 5.32i)11-s − 13.8i·12-s + (−5.90 + 11.5i)13-s − 5.32·14-s + (−14.9 + 14.9i)15-s − 8.35·16-s − 21.9i·17-s + ⋯
L(s)  = 1  + (0.290 − 0.290i)2-s − 1.38·3-s + 0.831i·4-s + (0.716 − 0.716i)5-s + (−0.403 + 0.403i)6-s + (−0.654 − 0.654i)7-s + (0.532 + 0.532i)8-s + 0.924·9-s − 0.416i·10-s + (0.484 + 0.484i)11-s − 1.15i·12-s + (−0.454 + 0.890i)13-s − 0.380·14-s + (−0.993 + 0.993i)15-s − 0.521·16-s − 1.29i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.984 + 0.176i$
motivic weight  =  \(2\)
character  :  $\chi_{13} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :1),\ 0.984 + 0.176i)$
$L(\frac{3}{2})$  $\approx$  $0.648809 - 0.0577555i$
$L(\frac12)$  $\approx$  $0.648809 - 0.0577555i$
$L(2)$   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 \( 1 + (5.90 - 11.5i)T \)
good2 \( 1 + (-0.581 + 0.581i)T - 4iT^{2} \)
3 \( 1 + 4.16T + 9T^{2} \)
5 \( 1 + (-3.58 + 3.58i)T - 25iT^{2} \)
7 \( 1 + (4.58 + 4.58i)T + 49iT^{2} \)
11 \( 1 + (-5.32 - 5.32i)T + 121iT^{2} \)
17 \( 1 + 21.9iT - 289T^{2} \)
19 \( 1 + (-3.16 + 3.16i)T - 361iT^{2} \)
23 \( 1 + 8.51iT - 529T^{2} \)
29 \( 1 + 5.81T + 841T^{2} \)
31 \( 1 + (-0.513 + 0.513i)T - 961iT^{2} \)
37 \( 1 + (-24.2 - 24.2i)T + 1.36e3iT^{2} \)
41 \( 1 + (-4.83 + 4.83i)T - 1.68e3iT^{2} \)
43 \( 1 - 30.4iT - 1.84e3T^{2} \)
47 \( 1 + (37.3 + 37.3i)T + 2.20e3iT^{2} \)
53 \( 1 + 35.8T + 2.80e3T^{2} \)
59 \( 1 + (-58.2 - 58.2i)T + 3.48e3iT^{2} \)
61 \( 1 + 80.3T + 3.72e3T^{2} \)
67 \( 1 + (-39.0 + 39.0i)T - 4.48e3iT^{2} \)
71 \( 1 + (-91.5 + 91.5i)T - 5.04e3iT^{2} \)
73 \( 1 + (-31.6 - 31.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 18.7T + 6.24e3T^{2} \)
83 \( 1 + (44.6 - 44.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (-8.89 - 8.89i)T + 7.92e3iT^{2} \)
97 \( 1 + (121. - 121. i)T - 9.40e3iT^{2} \)
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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

−20.07705563370172261207446866316, −17.95968323919624174885987371043, −16.81981330681481747967995918196, −16.51199170187262004701105122947, −13.67826964445310405211522983516, −12.48767843173433937554344724520, −11.43954033197981425354098322200, −9.563996724071608323299871578343, −6.85874272103586218025205582972, −4.77450417013775926355845063463, 5.65990377670370487912622172668, 6.38261627920717730506117029383, 9.896974242426991948391952809843, 10.99395777799112031275258853299, 12.73477724760375199072324025869, 14.45109489363855657383888504907, 15.76229596423075253929326431430, 17.18608481629438217751192576964, 18.33323662465042175742064478334, 19.49695197618202841717339595129

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