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Properties

Label 2-13-13.10-c9-0-0
Degree 22
Conductor 1313
Sign −0.218−0.975i-0.218 - 0.975i
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  + (−29.6 + 17.1i)2-s + (−132. − 228. i)3-s + (330. − 571. i)4-s − 402. i·5-s + (7.82e3 + 4.51e3i)6-s + (−8.11e3 − 4.68e3i)7-s + 5.06e3i·8-s + (−2.50e4 + 4.33e4i)9-s + (6.88e3 + 1.19e4i)10-s + (1.54e4 − 8.90e3i)11-s − 1.74e5·12-s + (7.06e4 + 7.49e4i)13-s + 3.20e5·14-s + (−9.19e4 + 5.31e4i)15-s + (8.22e4 + 1.42e5i)16-s + (2.32e4 − 4.03e4i)17-s + ⋯
L(s)  = 1  + (−1.31 + 0.756i)2-s + (−0.940 − 1.62i)3-s + (0.644 − 1.11i)4-s − 0.287i·5-s + (2.46 + 1.42i)6-s + (−1.27 − 0.737i)7-s + 0.437i·8-s + (−1.27 + 2.20i)9-s + (0.217 + 0.377i)10-s + (0.317 − 0.183i)11-s − 2.42·12-s + (0.686 + 0.727i)13-s + 2.23·14-s + (−0.469 + 0.270i)15-s + (0.313 + 0.543i)16-s + (0.0675 − 0.117i)17-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=((−0.218−0.975i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+9/2) L(s)=((−0.218−0.975i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: −0.218−0.975i-0.218 - 0.975i
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), −0.218−0.975i)(2,\ 13,\ (\ :9/2),\ -0.218 - 0.975i)

Particular Values

L(5)L(5) ≈\approx 0.0359908+0.0449199i0.0359908 + 0.0449199i
L(12)L(\frac12) ≈\approx 0.0359908+0.0449199i0.0359908 + 0.0449199i
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+(−7.06e4−7.49e4i)T 1 + (-7.06e4 - 7.49e4i)T
good2 1+(29.6−17.1i)T+(256−443.i)T2 1 + (29.6 - 17.1i)T + (256 - 443. i)T^{2}
3 1+(132.+228.i)T+(−9.84e3+1.70e4i)T2 1 + (132. + 228. i)T + (-9.84e3 + 1.70e4i)T^{2}
5 1+402.iT−1.95e6T2 1 + 402. iT - 1.95e6T^{2}
7 1+(8.11e3+4.68e3i)T+(2.01e7+3.49e7i)T2 1 + (8.11e3 + 4.68e3i)T + (2.01e7 + 3.49e7i)T^{2}
11 1+(−1.54e4+8.90e3i)T+(1.17e9−2.04e9i)T2 1 + (-1.54e4 + 8.90e3i)T + (1.17e9 - 2.04e9i)T^{2}
17 1+(−2.32e4+4.03e4i)T+(−5.92e10−1.02e11i)T2 1 + (-2.32e4 + 4.03e4i)T + (-5.92e10 - 1.02e11i)T^{2}
19 1+(2.51e5+1.45e5i)T+(1.61e11+2.79e11i)T2 1 + (2.51e5 + 1.45e5i)T + (1.61e11 + 2.79e11i)T^{2}
23 1+(8.16e5+1.41e6i)T+(−9.00e11+1.55e12i)T2 1 + (8.16e5 + 1.41e6i)T + (-9.00e11 + 1.55e12i)T^{2}
29 1+(1.75e5+3.04e5i)T+(−7.25e12+1.25e13i)T2 1 + (1.75e5 + 3.04e5i)T + (-7.25e12 + 1.25e13i)T^{2}
31 1−6.33e6iT−2.64e13T2 1 - 6.33e6iT - 2.64e13T^{2}
37 1+(1.06e7−6.13e6i)T+(6.49e13−1.12e14i)T2 1 + (1.06e7 - 6.13e6i)T + (6.49e13 - 1.12e14i)T^{2}
41 1+(2.52e6−1.45e6i)T+(1.63e14−2.83e14i)T2 1 + (2.52e6 - 1.45e6i)T + (1.63e14 - 2.83e14i)T^{2}
43 1+(2.83e6−4.90e6i)T+(−2.51e14−4.35e14i)T2 1 + (2.83e6 - 4.90e6i)T + (-2.51e14 - 4.35e14i)T^{2}
47 1+1.35e7iT−1.11e15T2 1 + 1.35e7iT - 1.11e15T^{2}
53 1−5.34e7T+3.29e15T2 1 - 5.34e7T + 3.29e15T^{2}
59 1+(8.14e7+4.70e7i)T+(4.33e15+7.50e15i)T2 1 + (8.14e7 + 4.70e7i)T + (4.33e15 + 7.50e15i)T^{2}
61 1+(7.82e7−1.35e8i)T+(−5.84e15−1.01e16i)T2 1 + (7.82e7 - 1.35e8i)T + (-5.84e15 - 1.01e16i)T^{2}
67 1+(1.50e8−8.71e7i)T+(1.36e16−2.35e16i)T2 1 + (1.50e8 - 8.71e7i)T + (1.36e16 - 2.35e16i)T^{2}
71 1+(9.79e6+5.65e6i)T+(2.29e16+3.97e16i)T2 1 + (9.79e6 + 5.65e6i)T + (2.29e16 + 3.97e16i)T^{2}
73 1+8.79e7iT−5.88e16T2 1 + 8.79e7iT - 5.88e16T^{2}
79 1−3.00e8T+1.19e17T2 1 - 3.00e8T + 1.19e17T^{2}
83 1−7.05e8iT−1.86e17T2 1 - 7.05e8iT - 1.86e17T^{2}
89 1+(4.64e8−2.68e8i)T+(1.75e17−3.03e17i)T2 1 + (4.64e8 - 2.68e8i)T + (1.75e17 - 3.03e17i)T^{2}
97 1+(−1.94e8−1.12e8i)T+(3.80e17+6.58e17i)T2 1 + (-1.94e8 - 1.12e8i)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

−18.03865138829707888978788380080, −16.76792020939973398655779558065, −16.37625818387532477987134223569, −13.61727289263345745862411726835, −12.40409790227170641348701802264, −10.58828017876137512725163920891, −8.620191658711198375596086383097, −6.97849885225450933537466177701, −6.36686800313681186165959894170, −1.12466446186515263852269366091, 0.06532031699282949588703967749, 3.35588321488004131645965664097, 5.88947573333368248204089138617, 9.007971184333308046999869904463, 9.934382566076870518194725668338, 10.88817123122628854886415716231, 12.16984685162834603591955314793, 15.23648408485991098899234634876, 16.26239877014758169029201210049, 17.32531430225017745395516626199

Graph of the ZZ-function along the critical line