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Properties

Label 2-13-13.12-c9-0-3
Degree 22
Conductor 1313
Sign −0.995−0.0949i-0.995 - 0.0949i
Analytic cond. 6.695466.69546
Root an. cond. 2.587552.58755
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 36.6i·2-s + 126.·3-s − 829.·4-s + 2.09e3i·5-s + 4.65e3i·6-s − 3.35e3i·7-s − 1.16e4i·8-s − 3.55e3·9-s − 7.67e4·10-s − 3.02e4i·11-s − 1.05e5·12-s + (1.02e5 + 9.78e3i)13-s + 1.22e5·14-s + 2.66e5i·15-s + 847.·16-s + 1.66e4·17-s + ⋯
L(s)  = 1  + 1.61i·2-s + 0.905·3-s − 1.61·4-s + 1.49i·5-s + 1.46i·6-s − 0.527i·7-s − 1.00i·8-s − 0.180·9-s − 2.42·10-s − 0.621i·11-s − 1.46·12-s + (0.995 + 0.0949i)13-s + 0.854·14-s + 1.35i·15-s + 0.00323·16-s + 0.0484·17-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=((−0.995−0.0949i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0949i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+9/2) L(s)=((−0.995−0.0949i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0949i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: −0.995−0.0949i-0.995 - 0.0949i
Analytic conductor: 6.695466.69546
Root analytic conductor: 2.587552.58755
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ13(12,⋅)\chi_{13} (12, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 13, ( :9/2), −0.995−0.0949i)(2,\ 13,\ (\ :9/2),\ -0.995 - 0.0949i)

Particular Values

L(5)L(5) ≈\approx 0.0890652+1.87087i0.0890652 + 1.87087i
L(12)L(\frac12) ≈\approx 0.0890652+1.87087i0.0890652 + 1.87087i
L(112)L(\frac{11}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad13 1+(−1.02e5−9.78e3i)T 1 + (-1.02e5 - 9.78e3i)T
good2 1−36.6iT−512T2 1 - 36.6iT - 512T^{2}
3 1−126.T+1.96e4T2 1 - 126.T + 1.96e4T^{2}
5 1−2.09e3iT−1.95e6T2 1 - 2.09e3iT - 1.95e6T^{2}
7 1+3.35e3iT−4.03e7T2 1 + 3.35e3iT - 4.03e7T^{2}
11 1+3.02e4iT−2.35e9T2 1 + 3.02e4iT - 2.35e9T^{2}
17 1−1.66e4T+1.18e11T2 1 - 1.66e4T + 1.18e11T^{2}
19 1−9.72e5iT−3.22e11T2 1 - 9.72e5iT - 3.22e11T^{2}
23 1−2.50e6T+1.80e12T2 1 - 2.50e6T + 1.80e12T^{2}
29 1−4.39e6T+1.45e13T2 1 - 4.39e6T + 1.45e13T^{2}
31 1−3.88e6iT−2.64e13T2 1 - 3.88e6iT - 2.64e13T^{2}
37 1+8.44e6iT−1.29e14T2 1 + 8.44e6iT - 1.29e14T^{2}
41 1+9.82e6iT−3.27e14T2 1 + 9.82e6iT - 3.27e14T^{2}
43 1−1.24e7T+5.02e14T2 1 - 1.24e7T + 5.02e14T^{2}
47 1+2.73e7iT−1.11e15T2 1 + 2.73e7iT - 1.11e15T^{2}
53 1+4.22e7T+3.29e15T2 1 + 4.22e7T + 3.29e15T^{2}
59 1−2.26e7iT−8.66e15T2 1 - 2.26e7iT - 8.66e15T^{2}
61 1+1.70e8T+1.16e16T2 1 + 1.70e8T + 1.16e16T^{2}
67 1+4.12e5iT−2.72e16T2 1 + 4.12e5iT - 2.72e16T^{2}
71 1+2.40e8iT−4.58e16T2 1 + 2.40e8iT - 4.58e16T^{2}
73 1+3.37e8iT−5.88e16T2 1 + 3.37e8iT - 5.88e16T^{2}
79 1+1.13e8T+1.19e17T2 1 + 1.13e8T + 1.19e17T^{2}
83 1−5.09e8iT−1.86e17T2 1 - 5.09e8iT - 1.86e17T^{2}
89 1+4.18e8iT−3.50e17T2 1 + 4.18e8iT - 3.50e17T^{2}
97 1+1.22e9iT−7.60e17T2 1 + 1.22e9iT - 7.60e17T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−18.20981248412863687796239164049, −16.75220495744961901038050847954, −15.36239809862658829406034735837, −14.32831984120264013039052101723, −13.80322834457947577244573513850, −10.76808135582743352297690246005, −8.711348364633175660673856509886, −7.43870275949815888705418467051, −6.13466625895213349378168069059, −3.37158820459940897695569841675, 1.05428497206120721649969871003, 2.77312506341140626402271146247, 4.68399068777890171775510588567, 8.668856699017719384478027268124, 9.303192833167714712302811850852, 11.33543253218959589326696697561, 12.71880655198529044444180864860, 13.50897546040615358085908378375, 15.49382792723207697686115102905, 17.41689722508960859283882390018

Graph of the ZZ-function along the critical line