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Properties

Label 2-13-13.5-c8-0-6
Degree $2$
Conductor $13$
Sign $0.997 - 0.0759i$
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  + (5.06 − 5.06i)2-s + 152.·3-s + 204. i·4-s + (−186. + 186. i)5-s + (773. − 773. i)6-s + (−798. − 798. i)7-s + (2.33e3 + 2.33e3i)8-s + 1.67e4·9-s + 1.88e3i·10-s + (−1.54e4 − 1.54e4i)11-s + 3.12e4i·12-s + (6.17e3 − 2.78e4i)13-s − 8.09e3·14-s + (−2.84e4 + 2.84e4i)15-s − 2.87e4·16-s − 2.39e3i·17-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)2-s + 1.88·3-s + 0.799i·4-s + (−0.297 + 0.297i)5-s + (0.596 − 0.596i)6-s + (−0.332 − 0.332i)7-s + (0.569 + 0.569i)8-s + 2.55·9-s + 0.188i·10-s + (−1.05 − 1.05i)11-s + 1.50i·12-s + (0.216 − 0.976i)13-s − 0.210·14-s + (−0.561 + 0.561i)15-s − 0.439·16-s − 0.0286i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0759i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.997 - 0.0759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.997 - 0.0759i$
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: \((2,\ 13,\ (\ :4),\ 0.997 - 0.0759i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.84000 + 0.108013i\)
\(L(\frac12)\) \(\approx\) \(2.84000 + 0.108013i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-6.17e3 + 2.78e4i)T \)
good2 \( 1 + (-5.06 + 5.06i)T - 256iT^{2} \)
3 \( 1 - 152.T + 6.56e3T^{2} \)
5 \( 1 + (186. - 186. i)T - 3.90e5iT^{2} \)
7 \( 1 + (798. + 798. i)T + 5.76e6iT^{2} \)
11 \( 1 + (1.54e4 + 1.54e4i)T + 2.14e8iT^{2} \)
17 \( 1 + 2.39e3iT - 6.97e9T^{2} \)
19 \( 1 + (1.14e5 - 1.14e5i)T - 1.69e10iT^{2} \)
23 \( 1 + 1.82e5iT - 7.83e10T^{2} \)
29 \( 1 + 2.26e5T + 5.00e11T^{2} \)
31 \( 1 + (-2.75e5 + 2.75e5i)T - 8.52e11iT^{2} \)
37 \( 1 + (2.45e5 + 2.45e5i)T + 3.51e12iT^{2} \)
41 \( 1 + (-4.92e5 + 4.92e5i)T - 7.98e12iT^{2} \)
43 \( 1 - 4.02e6iT - 1.16e13T^{2} \)
47 \( 1 + (-1.35e6 - 1.35e6i)T + 2.38e13iT^{2} \)
53 \( 1 - 1.23e7T + 6.22e13T^{2} \)
59 \( 1 + (-6.97e6 - 6.97e6i)T + 1.46e14iT^{2} \)
61 \( 1 + 9.38e6T + 1.91e14T^{2} \)
67 \( 1 + (1.59e7 - 1.59e7i)T - 4.06e14iT^{2} \)
71 \( 1 + (-3.34e7 + 3.34e7i)T - 6.45e14iT^{2} \)
73 \( 1 + (9.68e6 + 9.68e6i)T + 8.06e14iT^{2} \)
79 \( 1 + 5.01e7T + 1.51e15T^{2} \)
83 \( 1 + (2.44e7 - 2.44e7i)T - 2.25e15iT^{2} \)
89 \( 1 + (-5.38e7 - 5.38e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (3.06e7 - 3.06e7i)T - 7.83e15iT^{2} \)
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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

−18.47704490126334548652505530463, −16.29618784011241071183159437811, −14.98429251685472138369752709513, −13.57628632027882391433011404221, −12.86325423779856031522634356203, −10.52168624644099631746791809464, −8.511436650589731893473034150723, −7.65165055805375637192283610870, −3.73679552969547901523796167446, −2.71932879136412965571663166493, 2.15940522878395711979312276157, 4.41492354785326048150456053023, 7.15249774286876223887396272559, 8.831392300422416750315369175324, 10.06056101791047656632599904148, 12.88240405823813392349450627468, 13.94628960246226775458286657323, 15.18546603684696495917484068338, 15.76833812270271373685931079386, 18.52724308193263487986757600420

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