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Properties

Label 2-13-13.12-c13-0-7
Degree $2$
Conductor $13$
Sign $-0.285 + 0.958i$
Analytic cond. $13.9400$
Root an. cond. $3.73363$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.5i·2-s − 1.59e3·3-s + 8.08e3·4-s − 9.42e3i·5-s + 1.67e4i·6-s + 4.84e5i·7-s − 1.71e5i·8-s + 9.37e5·9-s − 9.91e4·10-s − 8.93e6i·11-s − 1.28e7·12-s + (4.96e6 − 1.66e7i)13-s + 5.10e6·14-s + 1.49e7i·15-s + 6.44e7·16-s − 1.57e8·17-s + ⋯
L(s)  = 1  − 0.116i·2-s − 1.26·3-s + 0.986·4-s − 0.269i·5-s + 0.146i·6-s + 1.55i·7-s − 0.230i·8-s + 0.587·9-s − 0.0313·10-s − 1.52i·11-s − 1.24·12-s + (0.285 − 0.958i)13-s + 0.181·14-s + 0.339i·15-s + 0.959·16-s − 1.58·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.285 + 0.958i$
Analytic conductor: \(13.9400\)
Root analytic conductor: \(3.73363\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :13/2),\ -0.285 + 0.958i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.568342 - 0.762176i\)
\(L(\frac12)\) \(\approx\) \(0.568342 - 0.762176i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-4.96e6 + 1.66e7i)T \)
good2 \( 1 + 10.5iT - 8.19e3T^{2} \)
3 \( 1 + 1.59e3T + 1.59e6T^{2} \)
5 \( 1 + 9.42e3iT - 1.22e9T^{2} \)
7 \( 1 - 4.84e5iT - 9.68e10T^{2} \)
11 \( 1 + 8.93e6iT - 3.45e13T^{2} \)
17 \( 1 + 1.57e8T + 9.90e15T^{2} \)
19 \( 1 + 3.07e8iT - 4.20e16T^{2} \)
23 \( 1 + 6.52e8T + 5.04e17T^{2} \)
29 \( 1 + 1.77e9T + 1.02e19T^{2} \)
31 \( 1 + 5.73e9iT - 2.44e19T^{2} \)
37 \( 1 - 4.96e9iT - 2.43e20T^{2} \)
41 \( 1 + 1.33e10iT - 9.25e20T^{2} \)
43 \( 1 - 3.47e10T + 1.71e21T^{2} \)
47 \( 1 + 5.71e10iT - 5.46e21T^{2} \)
53 \( 1 + 8.98e10T + 2.60e22T^{2} \)
59 \( 1 + 2.02e11iT - 1.04e23T^{2} \)
61 \( 1 - 9.24e10T + 1.61e23T^{2} \)
67 \( 1 + 2.80e11iT - 5.48e23T^{2} \)
71 \( 1 - 1.45e12iT - 1.16e24T^{2} \)
73 \( 1 - 6.04e11iT - 1.67e24T^{2} \)
79 \( 1 + 2.70e12T + 4.66e24T^{2} \)
83 \( 1 + 3.96e12iT - 8.87e24T^{2} \)
89 \( 1 - 5.41e12iT - 2.19e25T^{2} \)
97 \( 1 - 4.83e12iT - 6.73e25T^{2} \)
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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

−16.11169449161143819460165830525, −15.40421002424455797597136311801, −12.90515702011867163325415397521, −11.56921483382483921063314707990, −10.98520993526364886629008718284, −8.670798430829048633362953445643, −6.33790374056422136297733938490, −5.47992795255629289262123608003, −2.60238977262357075046700794591, −0.44574607887201194445777329516, 1.60832015164168544213010729871, 4.34663104488532979536245425827, 6.41554346171229817111750420393, 7.25831297021380377533017885711, 10.29712630773453535050886368360, 11.13920313679924114947052898850, 12.42454885034031591282657930292, 14.37375209438737307214547484777, 16.05844514672660794686207179546, 16.93150763555514198979834177841

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