Properties

Label 2-13-13.11-c4-0-1
Degree $2$
Conductor $13$
Sign $0.702 - 0.712i$
Analytic cond. $1.34380$
Root an. cond. $1.15922$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.387 + 1.44i)2-s + (3.29 + 5.70i)3-s + (11.9 + 6.87i)4-s + (−10.5 − 10.5i)5-s + (−9.52 + 2.55i)6-s + (−14.3 − 53.3i)7-s + (−31.4 + 31.4i)8-s + (18.8 − 32.5i)9-s + (19.3 − 11.1i)10-s + (−52.6 − 14.1i)11-s + 90.6i·12-s + (115. + 123. i)13-s + 82.7·14-s + (25.4 − 95.0i)15-s + (76.7 + 132. i)16-s + (−171. − 99.0i)17-s + ⋯
L(s)  = 1  + (−0.0968 + 0.361i)2-s + (0.365 + 0.633i)3-s + (0.744 + 0.429i)4-s + (−0.422 − 0.422i)5-s + (−0.264 + 0.0708i)6-s + (−0.291 − 1.08i)7-s + (−0.492 + 0.492i)8-s + (0.232 − 0.402i)9-s + (0.193 − 0.111i)10-s + (−0.435 − 0.116i)11-s + 0.629i·12-s + (0.684 + 0.729i)13-s + 0.421·14-s + (0.113 − 0.422i)15-s + (0.299 + 0.519i)16-s + (−0.593 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.712i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.702 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.702 - 0.712i$
Analytic conductor: \(1.34380\)
Root analytic conductor: \(1.15922\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :2),\ 0.702 - 0.712i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.12935 + 0.472428i\)
\(L(\frac12)\) \(\approx\) \(1.12935 + 0.472428i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-115. - 123. i)T \)
good2 \( 1 + (0.387 - 1.44i)T + (-13.8 - 8i)T^{2} \)
3 \( 1 + (-3.29 - 5.70i)T + (-40.5 + 70.1i)T^{2} \)
5 \( 1 + (10.5 + 10.5i)T + 625iT^{2} \)
7 \( 1 + (14.3 + 53.3i)T + (-2.07e3 + 1.20e3i)T^{2} \)
11 \( 1 + (52.6 + 14.1i)T + (1.26e4 + 7.32e3i)T^{2} \)
17 \( 1 + (171. + 99.0i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (106. - 28.5i)T + (1.12e5 - 6.51e4i)T^{2} \)
23 \( 1 + (761. - 439. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-398. - 690. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-1.10e3 - 1.10e3i)T + 9.23e5iT^{2} \)
37 \( 1 + (1.20e3 + 322. i)T + (1.62e6 + 9.37e5i)T^{2} \)
41 \( 1 + (-802. + 2.99e3i)T + (-2.44e6 - 1.41e6i)T^{2} \)
43 \( 1 + (798. + 461. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (995. - 995. i)T - 4.87e6iT^{2} \)
53 \( 1 - 3.86e3T + 7.89e6T^{2} \)
59 \( 1 + (-216. - 807. i)T + (-1.04e7 + 6.05e6i)T^{2} \)
61 \( 1 + (-1.38e3 + 2.39e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (783. - 2.92e3i)T + (-1.74e7 - 1.00e7i)T^{2} \)
71 \( 1 + (-4.08e3 + 1.09e3i)T + (2.20e7 - 1.27e7i)T^{2} \)
73 \( 1 + (918. - 918. i)T - 2.83e7iT^{2} \)
79 \( 1 + 7.90e3T + 3.89e7T^{2} \)
83 \( 1 + (-2.97e3 - 2.97e3i)T + 4.74e7iT^{2} \)
89 \( 1 + (-1.21e4 - 3.26e3i)T + (5.43e7 + 3.13e7i)T^{2} \)
97 \( 1 + (7.62e3 - 2.04e3i)T + (7.66e7 - 4.42e7i)T^{2} \)
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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

−19.76118869781080229605074818601, −17.69423442256621739076964638903, −16.16035099439651953865205196829, −15.76788827876822647948808156417, −13.94138561091911678216834884649, −12.12013767611598169614328069619, −10.45805030490539247627411922020, −8.559039382915840624199973215919, −6.86369860537749127240893058850, −3.88071576187802076261343566212, 2.46366264923141106715943862682, 6.28334529766259157779268461007, 8.110818512787731451654047049585, 10.30332323151845263401021695455, 11.77763062507697912819195651347, 13.14692736900172640435739909296, 15.07417218482899565828511795157, 15.89992406583644272335764744618, 18.28410556337792147455582376352, 19.02588351982128906408934075598

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