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Properties

Label 2-13-13.5-c2-0-0
Degree $2$
Conductor $13$
Sign $0.399 - 0.916i$
Analytic cond. $0.354224$
Root an. cond. $0.595167$
Motivic weight $2$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(13\)
Sign: $0.399 - 0.916i$
Analytic conductor: \(0.354224\)
Root analytic conductor: \(0.595167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :1),\ 0.399 - 0.916i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.424951 + 0.278524i\)
\(L(\frac12)\) \(\approx\) \(0.424951 + 0.278524i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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