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Properties

Label 2-13-13.5-c8-0-3
Degree $2$
Conductor $13$
Sign $0.979 + 0.202i$
Analytic cond. $5.29592$
Root an. cond. $2.30128$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−21.6 + 21.6i)2-s − 83.2·3-s − 679. i·4-s + (−706. + 706. i)5-s + (1.80e3 − 1.80e3i)6-s + (966. + 966. i)7-s + (9.15e3 + 9.15e3i)8-s + 376.·9-s − 3.05e4i·10-s + (−431. − 431. i)11-s + 5.65e4i·12-s + (1.36e4 − 2.50e4i)13-s − 4.17e4·14-s + (5.88e4 − 5.88e4i)15-s − 2.22e5·16-s − 5.32e3i·17-s + ⋯
L(s)  = 1  + (−1.35 + 1.35i)2-s − 1.02·3-s − 2.65i·4-s + (−1.12 + 1.12i)5-s + (1.38 − 1.38i)6-s + (0.402 + 0.402i)7-s + (2.23 + 2.23i)8-s + 0.0573·9-s − 3.05i·10-s + (−0.0294 − 0.0294i)11-s + 2.72i·12-s + (0.477 − 0.878i)13-s − 1.08·14-s + (1.16 − 1.16i)15-s − 3.39·16-s − 0.0638i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.979 + 0.202i$
Analytic conductor: \(5.29592\)
Root analytic conductor: \(2.30128\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :4),\ 0.979 + 0.202i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.164358 - 0.0168492i\)
\(L(\frac12)\) \(\approx\) \(0.164358 - 0.0168492i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.36e4 + 2.50e4i)T \)
good2 \( 1 + (21.6 - 21.6i)T - 256iT^{2} \)
3 \( 1 + 83.2T + 6.56e3T^{2} \)
5 \( 1 + (706. - 706. i)T - 3.90e5iT^{2} \)
7 \( 1 + (-966. - 966. i)T + 5.76e6iT^{2} \)
11 \( 1 + (431. + 431. i)T + 2.14e8iT^{2} \)
17 \( 1 + 5.32e3iT - 6.97e9T^{2} \)
19 \( 1 + (7.34e3 - 7.34e3i)T - 1.69e10iT^{2} \)
23 \( 1 + 8.77e4iT - 7.83e10T^{2} \)
29 \( 1 + 7.87e5T + 5.00e11T^{2} \)
31 \( 1 + (3.98e5 - 3.98e5i)T - 8.52e11iT^{2} \)
37 \( 1 + (-1.04e6 - 1.04e6i)T + 3.51e12iT^{2} \)
41 \( 1 + (-2.42e6 + 2.42e6i)T - 7.98e12iT^{2} \)
43 \( 1 + 1.12e6iT - 1.16e13T^{2} \)
47 \( 1 + (5.15e6 + 5.15e6i)T + 2.38e13iT^{2} \)
53 \( 1 - 1.30e7T + 6.22e13T^{2} \)
59 \( 1 + (6.77e6 + 6.77e6i)T + 1.46e14iT^{2} \)
61 \( 1 - 3.48e6T + 1.91e14T^{2} \)
67 \( 1 + (1.71e7 - 1.71e7i)T - 4.06e14iT^{2} \)
71 \( 1 + (-1.55e7 + 1.55e7i)T - 6.45e14iT^{2} \)
73 \( 1 + (-3.33e6 - 3.33e6i)T + 8.06e14iT^{2} \)
79 \( 1 + 4.95e6T + 1.51e15T^{2} \)
83 \( 1 + (3.26e7 - 3.26e7i)T - 2.25e15iT^{2} \)
89 \( 1 + (8.25e7 + 8.25e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (-7.02e7 + 7.02e7i)T - 7.83e15iT^{2} \)
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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

−17.97646669250885447796118300840, −16.64910831580664534831594324159, −15.50862920827233423846733760048, −14.66571386762146300379959235272, −11.41335709897208330064610733671, −10.51477836515437776812967821766, −8.344363586986005925737150367952, −7.04465947671759056733113790716, −5.64339827537401573434541712888, −0.22360960336736790852513505208, 1.08813289762842012708914415989, 4.20135765660155903914263465666, 7.78310320875123432077449412719, 9.119196316758428577541371328089, 11.07416839017323400299482729960, 11.65186317062690747023491861127, 12.80731011831451711262989053021, 16.30872142126931191051403073382, 16.87457701729625089746965549446, 18.09551165818044280444839977044

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