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Properties

Label 2-13-13.9-c3-0-1
Degree 22
Conductor 1313
Sign 0.973+0.230i0.973 + 0.230i
Analytic cond. 0.7670240.767024
Root an. cond. 0.8757990.875799
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.219 − 0.379i)2-s + (1.84 − 3.19i)3-s + (3.90 + 6.76i)4-s − 17.8·5-s + (−0.807 − 1.39i)6-s + (−2.71 − 4.70i)7-s + 6.93·8-s + (6.71 + 11.6i)9-s + (−3.90 + 6.76i)10-s + (11.2 − 19.4i)11-s + 28.7·12-s + (21.9 − 41.4i)13-s − 2.38·14-s + (−32.8 + 56.8i)15-s + (−29.7 + 51.4i)16-s + (−33.9 − 58.8i)17-s + ⋯
L(s)  = 1  + (0.0775 − 0.134i)2-s + (0.354 − 0.614i)3-s + (0.487 + 0.845i)4-s − 1.59·5-s + (−0.0549 − 0.0951i)6-s + (−0.146 − 0.254i)7-s + 0.306·8-s + (0.248 + 0.430i)9-s + (−0.123 + 0.213i)10-s + (0.307 − 0.532i)11-s + 0.692·12-s + (0.468 − 0.883i)13-s − 0.0455·14-s + (−0.564 + 0.978i)15-s + (−0.464 + 0.804i)16-s + (−0.484 − 0.839i)17-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=((0.973+0.230i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+3/2) L(s)=((0.973+0.230i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: 0.973+0.230i0.973 + 0.230i
Analytic conductor: 0.7670240.767024
Root analytic conductor: 0.8757990.875799
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ13(9,⋅)\chi_{13} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 13, ( :3/2), 0.973+0.230i)(2,\ 13,\ (\ :3/2),\ 0.973 + 0.230i)

Particular Values

L(2)L(2) ≈\approx 0.982027−0.114691i0.982027 - 0.114691i
L(12)L(\frac12) ≈\approx 0.982027−0.114691i0.982027 - 0.114691i
L(52)L(\frac{5}{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+(−21.9+41.4i)T 1 + (-21.9 + 41.4i)T
good2 1+(−0.219+0.379i)T+(−4−6.92i)T2 1 + (-0.219 + 0.379i)T + (-4 - 6.92i)T^{2}
3 1+(−1.84+3.19i)T+(−13.5−23.3i)T2 1 + (-1.84 + 3.19i)T + (-13.5 - 23.3i)T^{2}
5 1+17.8T+125T2 1 + 17.8T + 125T^{2}
7 1+(2.71+4.70i)T+(−171.5+297.i)T2 1 + (2.71 + 4.70i)T + (-171.5 + 297. i)T^{2}
11 1+(−11.2+19.4i)T+(−665.5−1.15e3i)T2 1 + (-11.2 + 19.4i)T + (-665.5 - 1.15e3i)T^{2}
17 1+(33.9+58.8i)T+(−2.45e3+4.25e3i)T2 1 + (33.9 + 58.8i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(−40.4−69.9i)T+(−3.42e3+5.94e3i)T2 1 + (-40.4 - 69.9i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(70.2−121.i)T+(−6.08e3−1.05e4i)T2 1 + (70.2 - 121. i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(−53.3+92.3i)T+(−1.21e4−2.11e4i)T2 1 + (-53.3 + 92.3i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+276.T+2.97e4T2 1 + 276.T + 2.97e4T^{2}
37 1+(−2.14+3.71i)T+(−2.53e4−4.38e4i)T2 1 + (-2.14 + 3.71i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1+(113.−197.i)T+(−3.44e4−5.96e4i)T2 1 + (113. - 197. i)T + (-3.44e4 - 5.96e4i)T^{2}
43 1+(13.7+23.8i)T+(−3.97e4+6.88e4i)T2 1 + (13.7 + 23.8i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1−318.T+1.03e5T2 1 - 318.T + 1.03e5T^{2}
53 1+67.6T+1.48e5T2 1 + 67.6T + 1.48e5T^{2}
59 1+(−145.−252.i)T+(−1.02e5+1.77e5i)T2 1 + (-145. - 252. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(331.+574.i)T+(−1.13e5+1.96e5i)T2 1 + (331. + 574. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(−212.+368.i)T+(−1.50e5−2.60e5i)T2 1 + (-212. + 368. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+(−76.4−132.i)T+(−1.78e5+3.09e5i)T2 1 + (-76.4 - 132. i)T + (-1.78e5 + 3.09e5i)T^{2}
73 1−117.T+3.89e5T2 1 - 117.T + 3.89e5T^{2}
79 1−202.T+4.93e5T2 1 - 202.T + 4.93e5T^{2}
83 1−336.T+5.71e5T2 1 - 336.T + 5.71e5T^{2}
89 1+(359.−621.i)T+(−3.52e5−6.10e5i)T2 1 + (359. - 621. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+(379.+657.i)T+(−4.56e5+7.90e5i)T2 1 + (379. + 657. i)T + (-4.56e5 + 7.90e5i)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

−19.67780105452462756970465910990, −18.35101931534678645970805260435, −16.44675986116577346436403731362, −15.60674160626015570317270278982, −13.55961242981874110454866407054, −12.24656201119802978584149472660, −11.12718488914698104803801456386, −8.162302933922337311779364217199, −7.34663469630280678944139989589, −3.58625571942394858965526178874, 4.19022483144951431422246083571, 6.90403086611684217568404125648, 9.009732338948342568151131608751, 10.80163656029455800316775363706, 12.14658068558979807099510010550, 14.55847462133591598399845749321, 15.44544303217164622985200394439, 16.18066007274649173202413446868, 18.56744771117491312836050583330, 19.74589105824024445259505976782

Graph of the ZZ-function along the critical line