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Properties

Label 2-13-13.12-c3-0-0
Degree $2$
Conductor $13$
Sign $0.554 - 0.832i$
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  + 3i·2-s − 3-s − 4-s − 9i·5-s − 3i·6-s − 15i·7-s + 21i·8-s − 26·9-s + 27·10-s + 48i·11-s + 12-s + (26 − 39i)13-s + 45·14-s + 9i·15-s − 71·16-s − 45·17-s + ⋯
L(s)  = 1  + 1.06i·2-s − 0.192·3-s − 0.125·4-s − 0.804i·5-s − 0.204i·6-s − 0.809i·7-s + 0.928i·8-s − 0.962·9-s + 0.853·10-s + 1.31i·11-s + 0.0240·12-s + (0.554 − 0.832i)13-s + 0.859·14-s + 0.154i·15-s − 1.10·16-s − 0.642·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.554 - 0.832i$
Analytic conductor: \(0.767024\)
Root analytic conductor: \(0.875799\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :3/2),\ 0.554 - 0.832i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.825202 + 0.441634i\)
\(L(\frac12)\) \(\approx\) \(0.825202 + 0.441634i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-26 + 39i)T \)
good2 \( 1 - 3iT - 8T^{2} \)
3 \( 1 + T + 27T^{2} \)
5 \( 1 + 9iT - 125T^{2} \)
7 \( 1 + 15iT - 343T^{2} \)
11 \( 1 - 48iT - 1.33e3T^{2} \)
17 \( 1 + 45T + 4.91e3T^{2} \)
19 \( 1 - 6iT - 6.85e3T^{2} \)
23 \( 1 - 162T + 1.21e4T^{2} \)
29 \( 1 + 144T + 2.43e4T^{2} \)
31 \( 1 - 264iT - 2.97e4T^{2} \)
37 \( 1 + 303iT - 5.06e4T^{2} \)
41 \( 1 + 192iT - 6.89e4T^{2} \)
43 \( 1 + 97T + 7.95e4T^{2} \)
47 \( 1 + 111iT - 1.03e5T^{2} \)
53 \( 1 + 414T + 1.48e5T^{2} \)
59 \( 1 + 522iT - 2.05e5T^{2} \)
61 \( 1 - 376T + 2.26e5T^{2} \)
67 \( 1 + 36iT - 3.00e5T^{2} \)
71 \( 1 - 357iT - 3.57e5T^{2} \)
73 \( 1 - 1.09e3iT - 3.89e5T^{2} \)
79 \( 1 + 830T + 4.93e5T^{2} \)
83 \( 1 + 438iT - 5.71e5T^{2} \)
89 \( 1 - 438iT - 7.04e5T^{2} \)
97 \( 1 + 852iT - 9.12e5T^{2} \)
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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

−20.07747087618661690760159125164, −17.62314566303340338455370062924, −17.00472259346844510464561559806, −15.74602187090146680512445425353, −14.46122281821318661474247024287, −12.82368120640982122209391662241, −10.96032656074757330025078934392, −8.704185250031597270937396060796, −7.09714183483057059409876765045, −5.18901415598640781945830316564, 2.95310855429547515318810034846, 6.30719010199337973804419825585, 8.983836264546810122151905930528, 11.04973796206346326445617763542, 11.51890229060460885251302265496, 13.40313563714491926733499154473, 15.05289148393711810595773703217, 16.63463237935173972028166248689, 18.56570016008740859489133774140, 19.12066855542506934627592696216

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