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Properties

Label 2-845-65.2-c1-0-53
Degree 22
Conductor 845845
Sign 0.659+0.751i0.659 + 0.751i
Analytic cond. 6.747356.74735
Root an. cond. 2.597562.59756
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  + (0.246 + 0.427i)2-s + (0.908 + 0.243i)3-s + (0.878 − 1.52i)4-s + (2.21 − 0.284i)5-s + (0.120 + 0.448i)6-s + (−3.18 − 1.83i)7-s + 1.85·8-s + (−1.83 − 1.05i)9-s + (0.669 + 0.878i)10-s + (0.177 − 0.664i)11-s + (1.16 − 1.16i)12-s − 1.81i·14-s + (2.08 + 0.281i)15-s + (−1.29 − 2.24i)16-s + (0.614 + 2.29i)17-s − 1.04i·18-s + ⋯
L(s)  = 1  + (0.174 + 0.302i)2-s + (0.524 + 0.140i)3-s + (0.439 − 0.760i)4-s + (0.991 − 0.127i)5-s + (0.0490 + 0.183i)6-s + (−1.20 − 0.694i)7-s + 0.655·8-s + (−0.610 − 0.352i)9-s + (0.211 + 0.277i)10-s + (0.0536 − 0.200i)11-s + (0.337 − 0.337i)12-s − 0.485i·14-s + (0.538 + 0.0726i)15-s + (−0.324 − 0.562i)16-s + (0.149 + 0.556i)17-s − 0.246i·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((0.659+0.751i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((0.659+0.751i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: 0.659+0.751i0.659 + 0.751i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(587,⋅)\chi_{845} (587, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), 0.659+0.751i)(2,\ 845,\ (\ :1/2),\ 0.659 + 0.751i)

Particular Values

L(1)L(1) ≈\approx 2.09241−0.948340i2.09241 - 0.948340i
L(12)L(\frac12) ≈\approx 2.09241−0.948340i2.09241 - 0.948340i
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)
bad5 1+(−2.21+0.284i)T 1 + (-2.21 + 0.284i)T
13 1 1
good2 1+(−0.246−0.427i)T+(−1+1.73i)T2 1 + (-0.246 - 0.427i)T + (-1 + 1.73i)T^{2}
3 1+(−0.908−0.243i)T+(2.59+1.5i)T2 1 + (-0.908 - 0.243i)T + (2.59 + 1.5i)T^{2}
7 1+(3.18+1.83i)T+(3.5+6.06i)T2 1 + (3.18 + 1.83i)T + (3.5 + 6.06i)T^{2}
11 1+(−0.177+0.664i)T+(−9.52−5.5i)T2 1 + (-0.177 + 0.664i)T + (-9.52 - 5.5i)T^{2}
17 1+(−0.614−2.29i)T+(−14.7+8.5i)T2 1 + (-0.614 - 2.29i)T + (-14.7 + 8.5i)T^{2}
19 1+(−5.29+1.41i)T+(16.4−9.5i)T2 1 + (-5.29 + 1.41i)T + (16.4 - 9.5i)T^{2}
23 1+(−0.350+1.30i)T+(−19.9−11.5i)T2 1 + (-0.350 + 1.30i)T + (-19.9 - 11.5i)T^{2}
29 1+(−8.24+4.75i)T+(14.5−25.1i)T2 1 + (-8.24 + 4.75i)T + (14.5 - 25.1i)T^{2}
31 1+(4.81−4.81i)T−31iT2 1 + (4.81 - 4.81i)T - 31iT^{2}
37 1+(1.58−0.917i)T+(18.5−32.0i)T2 1 + (1.58 - 0.917i)T + (18.5 - 32.0i)T^{2}
41 1+(−0.534−0.143i)T+(35.5+20.5i)T2 1 + (-0.534 - 0.143i)T + (35.5 + 20.5i)T^{2}
43 1+(2.09−0.560i)T+(37.2−21.5i)T2 1 + (2.09 - 0.560i)T + (37.2 - 21.5i)T^{2}
47 1−3.80iT−47T2 1 - 3.80iT - 47T^{2}
53 1+(2.47−2.47i)T−53iT2 1 + (2.47 - 2.47i)T - 53iT^{2}
59 1+(2.69+10.0i)T+(−51.0+29.5i)T2 1 + (2.69 + 10.0i)T + (-51.0 + 29.5i)T^{2}
61 1+(3.09−5.36i)T+(−30.5−52.8i)T2 1 + (3.09 - 5.36i)T + (-30.5 - 52.8i)T^{2}
67 1+(−6.12−10.6i)T+(−33.5+58.0i)T2 1 + (-6.12 - 10.6i)T + (-33.5 + 58.0i)T^{2}
71 1+(−1.73−6.47i)T+(−61.4+35.5i)T2 1 + (-1.73 - 6.47i)T + (-61.4 + 35.5i)T^{2}
73 1+3.37T+73T2 1 + 3.37T + 73T^{2}
79 1−3.12iT−79T2 1 - 3.12iT - 79T^{2}
83 1−2.13iT−83T2 1 - 2.13iT - 83T^{2}
89 1+(−3.26−0.874i)T+(77.0+44.5i)T2 1 + (-3.26 - 0.874i)T + (77.0 + 44.5i)T^{2}
97 1+(−3.53+6.12i)T+(−48.5−84.0i)T2 1 + (-3.53 + 6.12i)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

−9.955285949746645028373796542079, −9.461384011030403901841137861844, −8.535089679118406133972174784342, −7.27012573786140360814496302296, −6.42913022675565511219989367332, −5.93053406437799495310551816150, −4.89863075745007563579266592800, −3.47214347753233444809079878108, −2.56284746214491979484427029938, −1.01969916957812235343399182345, 1.93290957803353920335864744863, 2.90970569768107473376049398212, 3.33200059264854442046286386785, 5.06986644291770976462679231703, 5.99065080180027095452482020828, 6.88801478677472196352642805740, 7.72562959871180684557206864955, 8.797804974002115465450165766022, 9.397797881846258869942244118608, 10.21471997997349345648809398293

Graph of the ZZ-function along the critical line