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Properties

Label 2-13-13.5-c10-0-9
Degree $2$
Conductor $13$
Sign $-0.992 - 0.126i$
Analytic cond. $8.25964$
Root an. cond. $2.87395$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (40.2 − 40.2i)2-s − 209.·3-s − 2.20e3i·4-s + (−791. + 791. i)5-s + (−8.41e3 + 8.41e3i)6-s + (−1.18e4 − 1.18e4i)7-s + (−4.76e4 − 4.76e4i)8-s − 1.51e4·9-s + 6.36e4i·10-s + (1.08e5 + 1.08e5i)11-s + 4.62e5i·12-s + (1.51e5 − 3.38e5i)13-s − 9.48e5·14-s + (1.65e5 − 1.65e5i)15-s − 1.56e6·16-s − 2.32e6i·17-s + ⋯
L(s)  = 1  + (1.25 − 1.25i)2-s − 0.861·3-s − 2.15i·4-s + (−0.253 + 0.253i)5-s + (−1.08 + 1.08i)6-s + (−0.702 − 0.702i)7-s + (−1.45 − 1.45i)8-s − 0.257·9-s + 0.636i·10-s + (0.671 + 0.671i)11-s + 1.85i·12-s + (0.408 − 0.912i)13-s − 1.76·14-s + (0.218 − 0.218i)15-s − 1.49·16-s − 1.63i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.126i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.992 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.992 - 0.126i$
Analytic conductor: \(8.25964\)
Root analytic conductor: \(2.87395\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :5),\ -0.992 - 0.126i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.107927 + 1.70397i\)
\(L(\frac12)\) \(\approx\) \(0.107927 + 1.70397i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.51e5 + 3.38e5i)T \)
good2 \( 1 + (-40.2 + 40.2i)T - 1.02e3iT^{2} \)
3 \( 1 + 209.T + 5.90e4T^{2} \)
5 \( 1 + (791. - 791. i)T - 9.76e6iT^{2} \)
7 \( 1 + (1.18e4 + 1.18e4i)T + 2.82e8iT^{2} \)
11 \( 1 + (-1.08e5 - 1.08e5i)T + 2.59e10iT^{2} \)
17 \( 1 + 2.32e6iT - 2.01e12T^{2} \)
19 \( 1 + (5.21e5 - 5.21e5i)T - 6.13e12iT^{2} \)
23 \( 1 + 1.11e7iT - 4.14e13T^{2} \)
29 \( 1 - 3.79e6T + 4.20e14T^{2} \)
31 \( 1 + (-2.11e6 + 2.11e6i)T - 8.19e14iT^{2} \)
37 \( 1 + (-6.39e7 - 6.39e7i)T + 4.80e15iT^{2} \)
41 \( 1 + (1.61e8 - 1.61e8i)T - 1.34e16iT^{2} \)
43 \( 1 + 2.06e7iT - 2.16e16T^{2} \)
47 \( 1 + (-1.64e7 - 1.64e7i)T + 5.25e16iT^{2} \)
53 \( 1 + 9.68e7T + 1.74e17T^{2} \)
59 \( 1 + (-3.72e8 - 3.72e8i)T + 5.11e17iT^{2} \)
61 \( 1 + 1.55e8T + 7.13e17T^{2} \)
67 \( 1 + (-6.29e8 + 6.29e8i)T - 1.82e18iT^{2} \)
71 \( 1 + (-1.48e9 + 1.48e9i)T - 3.25e18iT^{2} \)
73 \( 1 + (2.34e9 + 2.34e9i)T + 4.29e18iT^{2} \)
79 \( 1 - 4.68e9T + 9.46e18T^{2} \)
83 \( 1 + (5.99e8 - 5.99e8i)T - 1.55e19iT^{2} \)
89 \( 1 + (4.09e9 + 4.09e9i)T + 3.11e19iT^{2} \)
97 \( 1 + (2.61e8 - 2.61e8i)T - 7.37e19iT^{2} \)
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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

−16.64207459161589733076730638205, −14.86683366948853211937048947635, −13.48582870785006027059218319212, −12.24127522060571997527456570717, −11.20131061877294328216147302485, −10.04415543579388177993129446354, −6.47423348143710511938462103098, −4.77770718058359836976900483277, −3.10344020934778798068824741696, −0.65550438550532964928990297554, 3.82442144048887026480603196789, 5.66828894739128560274328425422, 6.48851534423524129642022620256, 8.604917270254115591465493466714, 11.57043776609234379425368866300, 12.64826094959058092375324343664, 14.06990955764001149444507601464, 15.51164928520093275735036739643, 16.47002966117425348383189605441, 17.36289253117843963271142941776

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