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Properties

Label 2-13-13.5-c8-0-5
Degree $2$
Conductor $13$
Sign $0.613 - 0.789i$
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  + (−18.0 + 18.0i)2-s + 107.·3-s − 394. i·4-s + (706. − 706. i)5-s + (−1.94e3 + 1.94e3i)6-s + (1.70e3 + 1.70e3i)7-s + (2.48e3 + 2.48e3i)8-s + 5.10e3·9-s + 2.54e4i·10-s + (5.80e3 + 5.80e3i)11-s − 4.25e4i·12-s + (−1.64e4 − 2.33e4i)13-s − 6.13e4·14-s + (7.62e4 − 7.62e4i)15-s + 1.11e4·16-s + 1.34e5i·17-s + ⋯
L(s)  = 1  + (−1.12 + 1.12i)2-s + 1.33·3-s − 1.53i·4-s + (1.12 − 1.12i)5-s + (−1.50 + 1.50i)6-s + (0.708 + 0.708i)7-s + (0.607 + 0.607i)8-s + 0.777·9-s + 2.54i·10-s + (0.396 + 0.396i)11-s − 2.05i·12-s + (−0.577 − 0.816i)13-s − 1.59·14-s + (1.50 − 1.50i)15-s + 0.169·16-s + 1.60i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.613 - 0.789i$
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: \((2,\ 13,\ (\ :4),\ 0.613 - 0.789i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.42926 + 0.699157i\)
\(L(\frac12)\) \(\approx\) \(1.42926 + 0.699157i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (1.64e4 + 2.33e4i)T \)
good2 \( 1 + (18.0 - 18.0i)T - 256iT^{2} \)
3 \( 1 - 107.T + 6.56e3T^{2} \)
5 \( 1 + (-706. + 706. i)T - 3.90e5iT^{2} \)
7 \( 1 + (-1.70e3 - 1.70e3i)T + 5.76e6iT^{2} \)
11 \( 1 + (-5.80e3 - 5.80e3i)T + 2.14e8iT^{2} \)
17 \( 1 - 1.34e5iT - 6.97e9T^{2} \)
19 \( 1 + (-8.40e4 + 8.40e4i)T - 1.69e10iT^{2} \)
23 \( 1 - 1.36e5iT - 7.83e10T^{2} \)
29 \( 1 + 9.58e5T + 5.00e11T^{2} \)
31 \( 1 + (-4.84e5 + 4.84e5i)T - 8.52e11iT^{2} \)
37 \( 1 + (1.25e6 + 1.25e6i)T + 3.51e12iT^{2} \)
41 \( 1 + (-6.87e5 + 6.87e5i)T - 7.98e12iT^{2} \)
43 \( 1 - 3.17e6iT - 1.16e13T^{2} \)
47 \( 1 + (5.07e5 + 5.07e5i)T + 2.38e13iT^{2} \)
53 \( 1 + 8.75e6T + 6.22e13T^{2} \)
59 \( 1 + (3.35e6 + 3.35e6i)T + 1.46e14iT^{2} \)
61 \( 1 - 3.53e6T + 1.91e14T^{2} \)
67 \( 1 + (4.45e6 - 4.45e6i)T - 4.06e14iT^{2} \)
71 \( 1 + (7.29e6 - 7.29e6i)T - 6.45e14iT^{2} \)
73 \( 1 + (-8.69e6 - 8.69e6i)T + 8.06e14iT^{2} \)
79 \( 1 + 5.57e7T + 1.51e15T^{2} \)
83 \( 1 + (3.21e6 - 3.21e6i)T - 2.25e15iT^{2} \)
89 \( 1 + (-4.69e7 - 4.69e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (5.15e7 - 5.15e7i)T - 7.83e15iT^{2} \)
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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

−17.77174993571277046524094576100, −17.11239691685493250119329715315, −15.38966014487218704631921306914, −14.50934700675703699514463741748, −12.91396982916874913293579617005, −9.691598306669164140590494614273, −8.883341147580752170174250778763, −7.903033748989241864948623307978, −5.58473233013852574577806454238, −1.71755900986639415385785803681, 1.80055210888057831430773111256, 3.05408422724028792070986887905, 7.42648699095810123177632983250, 9.123610058887037068405412110525, 10.06129112783840584007040657168, 11.43691213199679625365896369086, 13.88707011397485672817600438749, 14.39597894964160026856507631059, 17.05547503966128306630524671413, 18.29165897690822935362234922928

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