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Properties

Label 2-42e2-9.4-c1-0-11
Degree 22
Conductor 17641764
Sign −0.954−0.299i-0.954 - 0.299i
Analytic cond. 14.085614.0856
Root an. cond. 3.753083.75308
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  + (0.899 + 1.47i)3-s + (−1.19 + 2.06i)5-s + (−1.38 + 2.66i)9-s + (1.12 + 1.95i)11-s + (−2.37 + 4.12i)13-s + (−4.12 + 0.0937i)15-s + 4.30·17-s + 8.59·19-s + (−0.664 + 1.15i)23-s + (−0.343 − 0.595i)25-s + (−5.18 + 0.353i)27-s + (−3.87 − 6.71i)29-s + (0.405 − 0.702i)31-s + (−1.87 + 3.43i)33-s − 4.63·37-s + ⋯
L(s)  = 1  + (0.519 + 0.854i)3-s + (−0.533 + 0.923i)5-s + (−0.460 + 0.887i)9-s + (0.340 + 0.590i)11-s + (−0.659 + 1.14i)13-s + (−1.06 + 0.0242i)15-s + 1.04·17-s + 1.97·19-s + (−0.138 + 0.240i)23-s + (−0.0687 − 0.119i)25-s + (−0.997 + 0.0680i)27-s + (−0.720 − 1.24i)29-s + (0.0727 − 0.126i)31-s + (−0.327 + 0.597i)33-s − 0.761·37-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=((−0.954−0.299i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s+1/2) L(s)=((−0.954−0.299i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: −0.954−0.299i-0.954 - 0.299i
Analytic conductor: 14.085614.0856
Root analytic conductor: 3.753083.75308
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1764(589,⋅)\chi_{1764} (589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :1/2), −0.954−0.299i)(2,\ 1764,\ (\ :1/2),\ -0.954 - 0.299i)

Particular Values

L(1)L(1) ≈\approx 1.5910672101.591067210
L(12)L(\frac12) ≈\approx 1.5910672101.591067210
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
3 1+(−0.899−1.47i)T 1 + (-0.899 - 1.47i)T
7 1 1
good5 1+(1.19−2.06i)T+(−2.5−4.33i)T2 1 + (1.19 - 2.06i)T + (-2.5 - 4.33i)T^{2}
11 1+(−1.12−1.95i)T+(−5.5+9.52i)T2 1 + (-1.12 - 1.95i)T + (-5.5 + 9.52i)T^{2}
13 1+(2.37−4.12i)T+(−6.5−11.2i)T2 1 + (2.37 - 4.12i)T + (-6.5 - 11.2i)T^{2}
17 1−4.30T+17T2 1 - 4.30T + 17T^{2}
19 1−8.59T+19T2 1 - 8.59T + 19T^{2}
23 1+(0.664−1.15i)T+(−11.5−19.9i)T2 1 + (0.664 - 1.15i)T + (-11.5 - 19.9i)T^{2}
29 1+(3.87+6.71i)T+(−14.5+25.1i)T2 1 + (3.87 + 6.71i)T + (-14.5 + 25.1i)T^{2}
31 1+(−0.405+0.702i)T+(−15.5−26.8i)T2 1 + (-0.405 + 0.702i)T + (-15.5 - 26.8i)T^{2}
37 1+4.63T+37T2 1 + 4.63T + 37T^{2}
41 1+(5.00−8.66i)T+(−20.5−35.5i)T2 1 + (5.00 - 8.66i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.74+3.01i)T+(−21.5+37.2i)T2 1 + (1.74 + 3.01i)T + (-21.5 + 37.2i)T^{2}
47 1+(2.18+3.78i)T+(−23.5+40.7i)T2 1 + (2.18 + 3.78i)T + (-23.5 + 40.7i)T^{2}
53 1+11.6T+53T2 1 + 11.6T + 53T^{2}
59 1+(2.40−4.16i)T+(−29.5−51.0i)T2 1 + (2.40 - 4.16i)T + (-29.5 - 51.0i)T^{2}
61 1+(0.575+0.997i)T+(−30.5+52.8i)T2 1 + (0.575 + 0.997i)T + (-30.5 + 52.8i)T^{2}
67 1+(−2.06+3.57i)T+(−33.5−58.0i)T2 1 + (-2.06 + 3.57i)T + (-33.5 - 58.0i)T^{2}
71 1+4.41T+71T2 1 + 4.41T + 71T^{2}
73 1−12.1T+73T2 1 - 12.1T + 73T^{2}
79 1+(−4.23−7.33i)T+(−39.5+68.4i)T2 1 + (-4.23 - 7.33i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.817+1.41i)T+(−41.5+71.8i)T2 1 + (0.817 + 1.41i)T + (-41.5 + 71.8i)T^{2}
89 1−6.34T+89T2 1 - 6.34T + 89T^{2}
97 1+(5.98+10.3i)T+(−48.5+84.0i)T2 1 + (5.98 + 10.3i)T + (-48.5 + 84.0i)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

−9.787779796843341147099755689853, −9.106232426065079840212520447728, −7.86632038881775686583076533848, −7.49384265291209321831410650814, −6.62886235806004223610277976367, −5.41547438286584084589029695745, −4.62375272758219644344984676403, −3.62505145891025749528231019118, −3.08874631378687095862182860803, −1.85395951234835193805083400945, 0.57864575001140377987201137754, 1.48339491347311379881444950711, 3.08751551629647152099020237609, 3.52805109951419290894252523989, 5.06628715720092684326323710880, 5.54341221172753516372684457339, 6.72071651821924502786381314161, 7.71419774647337601822774990662, 7.912794711489195619391823134720, 8.867147486039285350804636708741

Graph of the ZZ-function along the critical line