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Properties

Label 2-13-13.3-c9-0-0
Degree 22
Conductor 1313
Sign 0.756−0.653i0.756 - 0.653i
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  + (−19.5 − 33.8i)2-s + (−36.0 − 62.4i)3-s + (−509. + 882. i)4-s − 1.00e3·5-s + (−1.41e3 + 2.44e3i)6-s + (190. − 329. i)7-s + 1.98e4·8-s + (7.23e3 − 1.25e4i)9-s + (1.97e4 + 3.41e4i)10-s + (3.11e4 + 5.39e4i)11-s + 7.34e4·12-s + (−8.48e4 − 5.83e4i)13-s − 1.48e4·14-s + (3.63e4 + 6.29e4i)15-s + (−1.26e5 − 2.19e5i)16-s + (−2.08e5 + 3.61e5i)17-s + ⋯
L(s)  = 1  + (−0.864 − 1.49i)2-s + (−0.257 − 0.445i)3-s + (−0.994 + 1.72i)4-s − 0.720·5-s + (−0.444 + 0.770i)6-s + (0.0299 − 0.0518i)7-s + 1.71·8-s + (0.367 − 0.636i)9-s + (0.623 + 1.07i)10-s + (0.641 + 1.11i)11-s + 1.02·12-s + (−0.823 − 0.566i)13-s − 0.103·14-s + (0.185 + 0.321i)15-s + (−0.483 − 0.838i)16-s + (−0.605 + 1.04i)17-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=((0.756−0.653i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+9/2) L(s)=((0.756−0.653i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: 0.756−0.653i0.756 - 0.653i
Analytic conductor: 6.695466.69546
Root analytic conductor: 2.587552.58755
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ13(3,⋅)\chi_{13} (3, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 13, ( :9/2), 0.756−0.653i)(2,\ 13,\ (\ :9/2),\ 0.756 - 0.653i)

Particular Values

L(5)L(5) ≈\approx 0.149606+0.0556838i0.149606 + 0.0556838i
L(12)L(\frac12) ≈\approx 0.149606+0.0556838i0.149606 + 0.0556838i
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+(8.48e4+5.83e4i)T 1 + (8.48e4 + 5.83e4i)T
good2 1+(19.5+33.8i)T+(−256+443.i)T2 1 + (19.5 + 33.8i)T + (-256 + 443. i)T^{2}
3 1+(36.0+62.4i)T+(−9.84e3+1.70e4i)T2 1 + (36.0 + 62.4i)T + (-9.84e3 + 1.70e4i)T^{2}
5 1+1.00e3T+1.95e6T2 1 + 1.00e3T + 1.95e6T^{2}
7 1+(−190.+329.i)T+(−2.01e7−3.49e7i)T2 1 + (-190. + 329. i)T + (-2.01e7 - 3.49e7i)T^{2}
11 1+(−3.11e4−5.39e4i)T+(−1.17e9+2.04e9i)T2 1 + (-3.11e4 - 5.39e4i)T + (-1.17e9 + 2.04e9i)T^{2}
17 1+(2.08e5−3.61e5i)T+(−5.92e10−1.02e11i)T2 1 + (2.08e5 - 3.61e5i)T + (-5.92e10 - 1.02e11i)T^{2}
19 1+(1.18e5−2.05e5i)T+(−1.61e11−2.79e11i)T2 1 + (1.18e5 - 2.05e5i)T + (-1.61e11 - 2.79e11i)T^{2}
23 1+(−5.86e5−1.01e6i)T+(−9.00e11+1.55e12i)T2 1 + (-5.86e5 - 1.01e6i)T + (-9.00e11 + 1.55e12i)T^{2}
29 1+(−2.97e6−5.15e6i)T+(−7.25e12+1.25e13i)T2 1 + (-2.97e6 - 5.15e6i)T + (-7.25e12 + 1.25e13i)T^{2}
31 1+5.96e6T+2.64e13T2 1 + 5.96e6T + 2.64e13T^{2}
37 1+(8.42e6+1.45e7i)T+(−6.49e13+1.12e14i)T2 1 + (8.42e6 + 1.45e7i)T + (-6.49e13 + 1.12e14i)T^{2}
41 1+(6.53e6+1.13e7i)T+(−1.63e14+2.83e14i)T2 1 + (6.53e6 + 1.13e7i)T + (-1.63e14 + 2.83e14i)T^{2}
43 1+(−9.45e6+1.63e7i)T+(−2.51e14−4.35e14i)T2 1 + (-9.45e6 + 1.63e7i)T + (-2.51e14 - 4.35e14i)T^{2}
47 1+4.10e7T+1.11e15T2 1 + 4.10e7T + 1.11e15T^{2}
53 1+9.90e6T+3.29e15T2 1 + 9.90e6T + 3.29e15T^{2}
59 1+(4.79e7−8.30e7i)T+(−4.33e15−7.50e15i)T2 1 + (4.79e7 - 8.30e7i)T + (-4.33e15 - 7.50e15i)T^{2}
61 1+(1.00e8−1.74e8i)T+(−5.84e15−1.01e16i)T2 1 + (1.00e8 - 1.74e8i)T + (-5.84e15 - 1.01e16i)T^{2}
67 1+(−2.73e6−4.73e6i)T+(−1.36e16+2.35e16i)T2 1 + (-2.73e6 - 4.73e6i)T + (-1.36e16 + 2.35e16i)T^{2}
71 1+(−1.64e8+2.85e8i)T+(−2.29e16−3.97e16i)T2 1 + (-1.64e8 + 2.85e8i)T + (-2.29e16 - 3.97e16i)T^{2}
73 1−1.75e8T+5.88e16T2 1 - 1.75e8T + 5.88e16T^{2}
79 1+2.17e7T+1.19e17T2 1 + 2.17e7T + 1.19e17T^{2}
83 1+1.59e7T+1.86e17T2 1 + 1.59e7T + 1.86e17T^{2}
89 1+(2.38e8+4.12e8i)T+(−1.75e17+3.03e17i)T2 1 + (2.38e8 + 4.12e8i)T + (-1.75e17 + 3.03e17i)T^{2}
97 1+(3.12e8−5.41e8i)T+(−3.80e17−6.58e17i)T2 1 + (3.12e8 - 5.41e8i)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

−18.01837970894455201742489738300, −17.28929903049588589287654530510, −15.11491872280691135397491079404, −12.63544970288734926812134623887, −12.09816551508132502386464926854, −10.59415006618634737377722912041, −9.160991657259376740167531768935, −7.37876536259149094380643403645, −3.84218785750404691704185362770, −1.62599414104385827620483166558, 0.12456454890181482203260916588, 4.79885007841664451270097154684, 6.71487300838998607594956680918, 8.157228965365230069572082694520, 9.578217167163690641342343011357, 11.36234125638855753635771576048, 13.93495958730859788240508100918, 15.34487941882152089174964484724, 16.27264500832589703228054269683, 17.07128310954985086954591974874

Graph of the ZZ-function along the critical line