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Properties

Label 2-13-13.10-c9-0-7
Degree 22
Conductor 1313
Sign 0.736+0.676i0.736 + 0.676i
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  + (31.7 − 18.3i)2-s + (65.0 + 112. i)3-s + (417. − 722. i)4-s − 1.41e3i·5-s + (4.13e3 + 2.38e3i)6-s + (6.50e3 + 3.75e3i)7-s − 1.18e4i·8-s + (1.38e3 − 2.40e3i)9-s + (−2.60e4 − 4.50e4i)10-s + (−3.71e4 + 2.14e4i)11-s + 1.08e5·12-s + (−1.00e5 − 2.10e4i)13-s + 2.75e5·14-s + (1.59e5 − 9.21e4i)15-s + (−3.47e3 − 6.01e3i)16-s + (−2.67e5 + 4.63e5i)17-s + ⋯
L(s)  = 1  + (1.40 − 0.810i)2-s + (0.463 + 0.802i)3-s + (0.814 − 1.41i)4-s − 1.01i·5-s + (1.30 + 0.751i)6-s + (1.02 + 0.590i)7-s − 1.02i·8-s + (0.0705 − 0.122i)9-s + (−0.822 − 1.42i)10-s + (−0.764 + 0.441i)11-s + 1.51·12-s + (−0.978 − 0.204i)13-s + 1.91·14-s + (0.814 − 0.470i)15-s + (−0.0132 − 0.0229i)16-s + (−0.777 + 1.34i)17-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=((0.736+0.676i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+9/2) L(s)=((0.736+0.676i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: 0.736+0.676i0.736 + 0.676i
Analytic conductor: 6.695466.69546
Root analytic conductor: 2.587552.58755
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ13(10,⋅)\chi_{13} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 13, ( :9/2), 0.736+0.676i)(2,\ 13,\ (\ :9/2),\ 0.736 + 0.676i)

Particular Values

L(5)L(5) ≈\approx 3.52738−1.37292i3.52738 - 1.37292i
L(12)L(\frac12) ≈\approx 3.52738−1.37292i3.52738 - 1.37292i
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+(1.00e5+2.10e4i)T 1 + (1.00e5 + 2.10e4i)T
good2 1+(−31.7+18.3i)T+(256−443.i)T2 1 + (-31.7 + 18.3i)T + (256 - 443. i)T^{2}
3 1+(−65.0−112.i)T+(−9.84e3+1.70e4i)T2 1 + (-65.0 - 112. i)T + (-9.84e3 + 1.70e4i)T^{2}
5 1+1.41e3iT−1.95e6T2 1 + 1.41e3iT - 1.95e6T^{2}
7 1+(−6.50e3−3.75e3i)T+(2.01e7+3.49e7i)T2 1 + (-6.50e3 - 3.75e3i)T + (2.01e7 + 3.49e7i)T^{2}
11 1+(3.71e4−2.14e4i)T+(1.17e9−2.04e9i)T2 1 + (3.71e4 - 2.14e4i)T + (1.17e9 - 2.04e9i)T^{2}
17 1+(2.67e5−4.63e5i)T+(−5.92e10−1.02e11i)T2 1 + (2.67e5 - 4.63e5i)T + (-5.92e10 - 1.02e11i)T^{2}
19 1+(3.62e5+2.09e5i)T+(1.61e11+2.79e11i)T2 1 + (3.62e5 + 2.09e5i)T + (1.61e11 + 2.79e11i)T^{2}
23 1+(−1.47e5−2.55e5i)T+(−9.00e11+1.55e12i)T2 1 + (-1.47e5 - 2.55e5i)T + (-9.00e11 + 1.55e12i)T^{2}
29 1+(8.10e5+1.40e6i)T+(−7.25e12+1.25e13i)T2 1 + (8.10e5 + 1.40e6i)T + (-7.25e12 + 1.25e13i)T^{2}
31 1+3.25e6iT−2.64e13T2 1 + 3.25e6iT - 2.64e13T^{2}
37 1+(1.73e7−1.00e7i)T+(6.49e13−1.12e14i)T2 1 + (1.73e7 - 1.00e7i)T + (6.49e13 - 1.12e14i)T^{2}
41 1+(−2.98e7+1.72e7i)T+(1.63e14−2.83e14i)T2 1 + (-2.98e7 + 1.72e7i)T + (1.63e14 - 2.83e14i)T^{2}
43 1+(−1.56e6+2.71e6i)T+(−2.51e14−4.35e14i)T2 1 + (-1.56e6 + 2.71e6i)T + (-2.51e14 - 4.35e14i)T^{2}
47 1+5.82e7iT−1.11e15T2 1 + 5.82e7iT - 1.11e15T^{2}
53 1+2.43e7T+3.29e15T2 1 + 2.43e7T + 3.29e15T^{2}
59 1+(−1.40e8−8.09e7i)T+(4.33e15+7.50e15i)T2 1 + (-1.40e8 - 8.09e7i)T + (4.33e15 + 7.50e15i)T^{2}
61 1+(4.21e7−7.30e7i)T+(−5.84e15−1.01e16i)T2 1 + (4.21e7 - 7.30e7i)T + (-5.84e15 - 1.01e16i)T^{2}
67 1+(4.23e7−2.44e7i)T+(1.36e16−2.35e16i)T2 1 + (4.23e7 - 2.44e7i)T + (1.36e16 - 2.35e16i)T^{2}
71 1+(2.87e7+1.65e7i)T+(2.29e16+3.97e16i)T2 1 + (2.87e7 + 1.65e7i)T + (2.29e16 + 3.97e16i)T^{2}
73 1+2.22e8iT−5.88e16T2 1 + 2.22e8iT - 5.88e16T^{2}
79 1−1.28e7T+1.19e17T2 1 - 1.28e7T + 1.19e17T^{2}
83 1−1.32e8iT−1.86e17T2 1 - 1.32e8iT - 1.86e17T^{2}
89 1+(−5.06e8+2.92e8i)T+(1.75e17−3.03e17i)T2 1 + (-5.06e8 + 2.92e8i)T + (1.75e17 - 3.03e17i)T^{2}
97 1+(8.94e8+5.16e8i)T+(3.80e17+6.58e17i)T2 1 + (8.94e8 + 5.16e8i)T + (3.80e17 + 6.58e17i)T^{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

−17.44296895297923404621444827764, −15.36316546446503783258356104732, −14.81193958579167265795561898971, −13.08388232536301038383201091948, −12.10241278166952661636503337705, −10.47203279502703026251932058059, −8.644288618099399098112673100658, −5.20264544616574041824251221733, −4.26227499782834824633665324755, −2.15122543136600175696083174013, 2.63263028186040645115975940662, 4.81389102714685375285711352454, 6.90057935616463042835917462995, 7.72197537833508371550077393113, 10.96523936435327421845760502993, 12.75111558363915598668991043421, 14.06541599285912086404244099890, 14.45910062377295646349955358236, 16.03497371408126738152287761486, 17.80862464104291171783907509784

Graph of the ZZ-function along the critical line