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Properties

Label 2-13-13.12-c11-0-5
Degree $2$
Conductor $13$
Sign $0.551 + 0.833i$
Analytic cond. $9.98846$
Root an. cond. $3.16045$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 25.1i·2-s − 638.·3-s + 1.41e3·4-s + 9.28e3i·5-s + 1.60e4i·6-s − 2.52e4i·7-s − 8.71e4i·8-s + 2.30e5·9-s + 2.33e5·10-s − 8.09e5i·11-s − 9.03e5·12-s + (7.38e5 + 1.11e6i)13-s − 6.34e5·14-s − 5.93e6i·15-s + 7.07e5·16-s + 1.09e7·17-s + ⋯
L(s)  = 1  − 0.555i·2-s − 1.51·3-s + 0.691·4-s + 1.32i·5-s + 0.843i·6-s − 0.567i·7-s − 0.939i·8-s + 1.30·9-s + 0.738·10-s − 1.51i·11-s − 1.04·12-s + (0.551 + 0.833i)13-s − 0.315·14-s − 2.01i·15-s + 0.168·16-s + 1.87·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.833i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.551 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.551 + 0.833i$
Analytic conductor: \(9.98846\)
Root analytic conductor: \(3.16045\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :11/2),\ 0.551 + 0.833i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.12454 - 0.604312i\)
\(L(\frac12)\) \(\approx\) \(1.12454 - 0.604312i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-7.38e5 - 1.11e6i)T \)
good2 \( 1 + 25.1iT - 2.04e3T^{2} \)
3 \( 1 + 638.T + 1.77e5T^{2} \)
5 \( 1 - 9.28e3iT - 4.88e7T^{2} \)
7 \( 1 + 2.52e4iT - 1.97e9T^{2} \)
11 \( 1 + 8.09e5iT - 2.85e11T^{2} \)
17 \( 1 - 1.09e7T + 3.42e13T^{2} \)
19 \( 1 + 1.08e7iT - 1.16e14T^{2} \)
23 \( 1 + 5.65e6T + 9.52e14T^{2} \)
29 \( 1 - 7.64e7T + 1.22e16T^{2} \)
31 \( 1 - 1.22e8iT - 2.54e16T^{2} \)
37 \( 1 + 5.45e7iT - 1.77e17T^{2} \)
41 \( 1 + 9.95e8iT - 5.50e17T^{2} \)
43 \( 1 + 3.96e8T + 9.29e17T^{2} \)
47 \( 1 - 7.01e8iT - 2.47e18T^{2} \)
53 \( 1 - 6.66e8T + 9.26e18T^{2} \)
59 \( 1 + 2.42e9iT - 3.01e19T^{2} \)
61 \( 1 + 8.91e9T + 4.35e19T^{2} \)
67 \( 1 - 6.44e9iT - 1.22e20T^{2} \)
71 \( 1 + 1.38e10iT - 2.31e20T^{2} \)
73 \( 1 - 1.48e10iT - 3.13e20T^{2} \)
79 \( 1 - 2.96e10T + 7.47e20T^{2} \)
83 \( 1 - 1.25e10iT - 1.28e21T^{2} \)
89 \( 1 + 9.03e10iT - 2.77e21T^{2} \)
97 \( 1 - 5.24e10iT - 7.15e21T^{2} \)
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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

−16.85707809803685061168196369658, −15.95028927555336104768387637728, −14.01824055831178442865610724904, −11.97415034434512948630544640310, −11.03995666380438300238820070942, −10.43850869574018190031743747656, −7.02881726781072038277187925974, −6.00461776131628974997139026487, −3.31376408730779791119978360364, −0.881357948941809491320370231078, 1.26856613898476074537150121438, 5.07712789387999113282216955217, 5.98702089387829585337725467372, 7.87767562116656553467604870153, 10.14876109729369904378551672036, 11.97640185640651298529589327730, 12.46745759306134676383030733385, 15.13664322180748992650171293610, 16.35549974221503022005704461625, 16.95703274166776758637284592430

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