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Properties

Label 2-836-11.4-c1-0-14
Degree 22
Conductor 836836
Sign 0.530+0.847i0.530 + 0.847i
Analytic cond. 6.675496.67549
Root an. cond. 2.583692.58369
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.145 + 0.448i)3-s + (2.86 − 2.07i)5-s + (−0.710 − 2.18i)7-s + (2.24 + 1.63i)9-s + (0.309 − 3.30i)11-s + (−0.616 − 0.448i)13-s + (0.515 + 1.58i)15-s + (−0.815 + 0.592i)17-s + (0.309 − 0.951i)19-s + 1.08·21-s − 2.94·23-s + (2.32 − 7.14i)25-s + (−2.20 + 1.60i)27-s + (−1.55 − 4.79i)29-s + (−2.67 − 1.94i)31-s + ⋯
L(s)  = 1  + (−0.0840 + 0.258i)3-s + (1.27 − 0.929i)5-s + (−0.268 − 0.826i)7-s + (0.749 + 0.544i)9-s + (0.0931 − 0.995i)11-s + (−0.171 − 0.124i)13-s + (0.132 + 0.409i)15-s + (−0.197 + 0.143i)17-s + (0.0708 − 0.218i)19-s + 0.236·21-s − 0.615·23-s + (0.464 − 1.42i)25-s + (−0.423 + 0.307i)27-s + (−0.289 − 0.890i)29-s + (−0.480 − 0.348i)31-s + ⋯

Functional equation

Λ(s)=(836s/2 ΓC(s) L(s)=((0.530+0.847i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(836s/2 ΓC(s+1/2) L(s)=((0.530+0.847i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 836836    =    22⋅11⋅192^{2} \cdot 11 \cdot 19
Sign: 0.530+0.847i0.530 + 0.847i
Analytic conductor: 6.675496.67549
Root analytic conductor: 2.583692.58369
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ836(609,⋅)\chi_{836} (609, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 836, ( :1/2), 0.530+0.847i)(2,\ 836,\ (\ :1/2),\ 0.530 + 0.847i)

Particular Values

L(1)L(1) ≈\approx 1.61962−0.897478i1.61962 - 0.897478i
L(12)L(\frac12) ≈\approx 1.61962−0.897478i1.61962 - 0.897478i
L(32)L(\frac{3}{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)
bad2 1 1
11 1+(−0.309+3.30i)T 1 + (-0.309 + 3.30i)T
19 1+(−0.309+0.951i)T 1 + (-0.309 + 0.951i)T
good3 1+(0.145−0.448i)T+(−2.42−1.76i)T2 1 + (0.145 - 0.448i)T + (-2.42 - 1.76i)T^{2}
5 1+(−2.86+2.07i)T+(1.54−4.75i)T2 1 + (-2.86 + 2.07i)T + (1.54 - 4.75i)T^{2}
7 1+(0.710+2.18i)T+(−5.66+4.11i)T2 1 + (0.710 + 2.18i)T + (-5.66 + 4.11i)T^{2}
13 1+(0.616+0.448i)T+(4.01+12.3i)T2 1 + (0.616 + 0.448i)T + (4.01 + 12.3i)T^{2}
17 1+(0.815−0.592i)T+(5.25−16.1i)T2 1 + (0.815 - 0.592i)T + (5.25 - 16.1i)T^{2}
23 1+2.94T+23T2 1 + 2.94T + 23T^{2}
29 1+(1.55+4.79i)T+(−23.4+17.0i)T2 1 + (1.55 + 4.79i)T + (-23.4 + 17.0i)T^{2}
31 1+(2.67+1.94i)T+(9.57+29.4i)T2 1 + (2.67 + 1.94i)T + (9.57 + 29.4i)T^{2}
37 1+(−2.40−7.39i)T+(−29.9+21.7i)T2 1 + (-2.40 - 7.39i)T + (-29.9 + 21.7i)T^{2}
41 1+(0.418−1.28i)T+(−33.1−24.0i)T2 1 + (0.418 - 1.28i)T + (-33.1 - 24.0i)T^{2}
43 1−3.91T+43T2 1 - 3.91T + 43T^{2}
47 1+(−0.146+0.451i)T+(−38.0−27.6i)T2 1 + (-0.146 + 0.451i)T + (-38.0 - 27.6i)T^{2}
53 1+(−4.35−3.16i)T+(16.3+50.4i)T2 1 + (-4.35 - 3.16i)T + (16.3 + 50.4i)T^{2}
59 1+(0.583+1.79i)T+(−47.7+34.6i)T2 1 + (0.583 + 1.79i)T + (-47.7 + 34.6i)T^{2}
61 1+(−2.42+1.75i)T+(18.8−58.0i)T2 1 + (-2.42 + 1.75i)T + (18.8 - 58.0i)T^{2}
67 1−12.5T+67T2 1 - 12.5T + 67T^{2}
71 1+(4.73−3.43i)T+(21.9−67.5i)T2 1 + (4.73 - 3.43i)T + (21.9 - 67.5i)T^{2}
73 1+(−2.49−7.67i)T+(−59.0+42.9i)T2 1 + (-2.49 - 7.67i)T + (-59.0 + 42.9i)T^{2}
79 1+(9.40+6.83i)T+(24.4+75.1i)T2 1 + (9.40 + 6.83i)T + (24.4 + 75.1i)T^{2}
83 1+(−10.8+7.84i)T+(25.6−78.9i)T2 1 + (-10.8 + 7.84i)T + (25.6 - 78.9i)T^{2}
89 1−12.0T+89T2 1 - 12.0T + 89T^{2}
97 1+(−2.58−1.87i)T+(29.9+92.2i)T2 1 + (-2.58 - 1.87i)T + (29.9 + 92.2i)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

−10.00327526890731176243654193948, −9.428799009269080287185311188413, −8.488975636934770294176051067922, −7.56333609995317408582364701891, −6.42417376653456100918271491364, −5.63959228145416225990540902570, −4.74506932855823632096807300775, −3.81452835039509228310541326748, −2.22685213673611190361083605786, −0.970764598044019689316380211212, 1.77559611553381443079806086878, 2.53091632548320361124809756900, 3.87918273010408139749016658491, 5.27065439223754216331396189309, 6.09703667410294358478922755817, 6.82428051459903991430194642209, 7.47406230762899728226728069640, 8.992274293320837867874738830446, 9.572056214582184544784817357363, 10.16890951115427515339024778172

Graph of the ZZ-function along the critical line