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Properties

Label 2-13-13.3-c3-0-2
Degree $2$
Conductor $13$
Sign $-0.0128 + 0.999i$
Analytic cond. $0.767024$
Root an. cond. $0.875799$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 3.46i)2-s + (−1 − 1.73i)3-s + (−3.99 + 6.92i)4-s + 17·5-s + (−3.99 + 6.92i)6-s + (−10 + 17.3i)7-s + (11.5 − 19.9i)9-s + (−34 − 58.8i)10-s + (16 + 27.7i)11-s + 15.9·12-s + (−45.5 + 11.2i)13-s + 80·14-s + (−17 − 29.4i)15-s + (31.9 + 55.4i)16-s + (6.5 − 11.2i)17-s − 92·18-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.192 − 0.333i)3-s + (−0.499 + 0.866i)4-s + 1.52·5-s + (−0.272 + 0.471i)6-s + (−0.539 + 0.935i)7-s + (0.425 − 0.737i)9-s + (−1.07 − 1.86i)10-s + (0.438 + 0.759i)11-s + 0.384·12-s + (−0.970 + 0.240i)13-s + 1.52·14-s + (−0.292 − 0.506i)15-s + (0.499 + 0.866i)16-s + (0.0927 − 0.160i)17-s − 1.20·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.0128 + 0.999i$
Analytic conductor: \(0.767024\)
Root analytic conductor: \(0.875799\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :3/2),\ -0.0128 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.523757 - 0.530517i\)
\(L(\frac12)\) \(\approx\) \(0.523757 - 0.530517i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (45.5 - 11.2i)T \)
good2 \( 1 + (2 + 3.46i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (1 + 1.73i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 17T + 125T^{2} \)
7 \( 1 + (10 - 17.3i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-16 - 27.7i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-6.5 + 11.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (15 - 25.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (39 + 67.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (98.5 + 170. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 74T + 2.97e4T^{2} \)
37 \( 1 + (-113.5 - 196. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-82.5 - 142. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-78 + 135. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 162T + 1.03e5T^{2} \)
53 \( 1 - 93T + 1.48e5T^{2} \)
59 \( 1 + (-432 + 748. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (72.5 - 125. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (431 + 746. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (327 - 566. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 215T + 3.89e5T^{2} \)
79 \( 1 + 76T + 4.93e5T^{2} \)
83 \( 1 - 628T + 5.71e5T^{2} \)
89 \( 1 + (-133 - 230. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (119 - 206. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−18.96574936302645102981319399967, −18.09254921048204300184648378740, −17.18157593915280664305678696405, −14.86177173843490193203706774909, −12.86927017481398855429709189815, −11.98747629761118495658865541461, −9.887502046509861969066788993308, −9.370262842225942902263192995265, −6.23548438188844542950081286302, −2.13695132026179111184784576896, 5.68016097036963500634962429051, 7.23055185159361865807648753128, 9.320592364977432349566439380192, 10.38286862212767469402173284607, 13.26329777588852698611397520397, 14.48917747574907568308876279142, 16.31112134567384953240925394430, 16.92124724524974774626837997417, 17.90997346769399799405801989332, 19.48087388305308515481482611374

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