Properties

Degree 2
Conductor 13
Sign $0.399 + 0.916i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.58 − 2.58i)2-s + 2.16·3-s + 9.32i·4-s + (0.418 + 0.418i)5-s + (−5.58 − 5.58i)6-s + (−1.41 + 1.41i)7-s + (13.7 − 13.7i)8-s − 4.32·9-s − 2.16i·10-s + (−7.32 + 7.32i)11-s + 20.1i·12-s + (9.90 − 8.41i)13-s + 7.32·14-s + (0.905 + 0.905i)15-s − 33.6·16-s − 15.9i·17-s + ⋯
L(s)  = 1  + (−1.29 − 1.29i)2-s + 0.720·3-s + 2.33i·4-s + (0.0837 + 0.0837i)5-s + (−0.930 − 0.930i)6-s + (−0.202 + 0.202i)7-s + (1.71 − 1.71i)8-s − 0.480·9-s − 0.216i·10-s + (−0.665 + 0.665i)11-s + 1.68i·12-s + (0.761 − 0.647i)13-s + 0.523·14-s + (0.0603 + 0.0603i)15-s − 2.10·16-s − 0.939i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.399 + 0.916i$
motivic weight  =  \(2\)
character  :  $\chi_{13} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :1),\ 0.399 + 0.916i)$
$L(\frac{3}{2})$  $\approx$  $0.424951 - 0.278524i$
$L(\frac12)$  $\approx$  $0.424951 - 0.278524i$
$L(2)$   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 \( 1 + (-9.90 + 8.41i)T \)
good2 \( 1 + (2.58 + 2.58i)T + 4iT^{2} \)
3 \( 1 - 2.16T + 9T^{2} \)
5 \( 1 + (-0.418 - 0.418i)T + 25iT^{2} \)
7 \( 1 + (1.41 - 1.41i)T - 49iT^{2} \)
11 \( 1 + (7.32 - 7.32i)T - 121iT^{2} \)
17 \( 1 + 15.9iT - 289T^{2} \)
19 \( 1 + (3.16 + 3.16i)T + 361iT^{2} \)
23 \( 1 - 27.4iT - 529T^{2} \)
29 \( 1 - 25.8T + 841T^{2} \)
31 \( 1 + (-19.4 - 19.4i)T + 961iT^{2} \)
37 \( 1 + (4.23 - 4.23i)T - 1.36e3iT^{2} \)
41 \( 1 + (-11.1 - 11.1i)T + 1.68e3iT^{2} \)
43 \( 1 + 11.5iT - 1.84e3T^{2} \)
47 \( 1 + (-35.3 + 35.3i)T - 2.20e3iT^{2} \)
53 \( 1 + 4.18T + 2.80e3T^{2} \)
59 \( 1 + (30.2 - 30.2i)T - 3.48e3iT^{2} \)
61 \( 1 + 67.6T + 3.72e3T^{2} \)
67 \( 1 + (81.0 + 81.0i)T + 4.48e3iT^{2} \)
71 \( 1 + (-50.4 - 50.4i)T + 5.04e3iT^{2} \)
73 \( 1 + (31.6 - 31.6i)T - 5.32e3iT^{2} \)
79 \( 1 - 50.7T + 6.24e3T^{2} \)
83 \( 1 + (-18.6 - 18.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (-91.1 + 91.1i)T - 7.92e3iT^{2} \)
97 \( 1 + (-87.3 - 87.3i)T + 9.40e3iT^{2} \)
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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

−19.69911499565106552817912694077, −18.41613043880210366164494295136, −17.51135788379122017091526354026, −15.73267954907706978465117940994, −13.56268633963231116245228884102, −12.03463667937509193806138129804, −10.50876836275333162196408802027, −9.177307123611610263205322290920, −7.921204180954394322383080744709, −2.85218477777242428197479268848, 6.16404177135625374340439059988, 8.056957803555827474452859535269, 8.998398593171744386351339134601, 10.65636187705288183345272844949, 13.69557914963411258049827622329, 14.90831937210881891886498767921, 16.17479581892520589519357467793, 17.20522381704729022489342949945, 18.65606451794125737962648612962, 19.45841386238803546328531060040

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