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Properties

Label 2-845-65.37-c1-0-9
Degree 22
Conductor 845845
Sign −0.905+0.424i-0.905 + 0.424i
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  + (−0.427 + 0.246i)2-s + (0.908 + 0.243i)3-s + (−0.878 + 1.52i)4-s + (0.284 + 2.21i)5-s + (−0.448 + 0.120i)6-s + (−1.83 + 3.18i)7-s − 1.85i·8-s + (−1.83 − 1.05i)9-s + (−0.669 − 0.878i)10-s + (−0.664 − 0.177i)11-s + (−1.16 + 1.16i)12-s − 1.81i·14-s + (−0.281 + 2.08i)15-s + (−1.29 − 2.24i)16-s + (−0.614 − 2.29i)17-s + 1.04·18-s + ⋯
L(s)  = 1  + (−0.302 + 0.174i)2-s + (0.524 + 0.140i)3-s + (−0.439 + 0.760i)4-s + (0.127 + 0.991i)5-s + (−0.183 + 0.0490i)6-s + (−0.694 + 1.20i)7-s − 0.655i·8-s + (−0.610 − 0.352i)9-s + (−0.211 − 0.277i)10-s + (−0.200 − 0.0536i)11-s + (−0.337 + 0.337i)12-s − 0.485i·14-s + (−0.0726 + 0.538i)15-s + (−0.324 − 0.562i)16-s + (−0.149 − 0.556i)17-s + 0.246·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((−0.905+0.424i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((−0.905+0.424i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: −0.905+0.424i-0.905 + 0.424i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(427,⋅)\chi_{845} (427, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), −0.905+0.424i)(2,\ 845,\ (\ :1/2),\ -0.905 + 0.424i)

Particular Values

L(1)L(1) ≈\approx 0.141785−0.636000i0.141785 - 0.636000i
L(12)L(\frac12) ≈\approx 0.141785−0.636000i0.141785 - 0.636000i
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+(−0.284−2.21i)T 1 + (-0.284 - 2.21i)T
13 1 1
good2 1+(0.427−0.246i)T+(1−1.73i)T2 1 + (0.427 - 0.246i)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+(1.83−3.18i)T+(−3.5−6.06i)T2 1 + (1.83 - 3.18i)T + (-3.5 - 6.06i)T^{2}
11 1+(0.664+0.177i)T+(9.52+5.5i)T2 1 + (0.664 + 0.177i)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+(−1.41−5.29i)T+(−16.4+9.5i)T2 1 + (-1.41 - 5.29i)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+(−0.917−1.58i)T+(−18.5+32.0i)T2 1 + (-0.917 - 1.58i)T + (-18.5 + 32.0i)T^{2}
41 1+(0.143−0.534i)T+(−35.5−20.5i)T2 1 + (0.143 - 0.534i)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 1−3.80T+47T2 1 - 3.80T + 47T^{2}
53 1+(2.47−2.47i)T−53iT2 1 + (2.47 - 2.47i)T - 53iT^{2}
59 1+(10.0−2.69i)T+(51.0−29.5i)T2 1 + (10.0 - 2.69i)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+(10.6−6.12i)T+(33.5−58.0i)T2 1 + (10.6 - 6.12i)T + (33.5 - 58.0i)T^{2}
71 1+(6.47−1.73i)T+(61.4−35.5i)T2 1 + (6.47 - 1.73i)T + (61.4 - 35.5i)T^{2}
73 1−3.37iT−73T2 1 - 3.37iT - 73T^{2}
79 1−3.12iT−79T2 1 - 3.12iT - 79T^{2}
83 1+2.13T+83T2 1 + 2.13T + 83T^{2}
89 1+(−0.874+3.26i)T+(−77.0−44.5i)T2 1 + (-0.874 + 3.26i)T + (-77.0 - 44.5i)T^{2}
97 1+(−6.12−3.53i)T+(48.5+84.0i)T2 1 + (-6.12 - 3.53i)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

−10.39599789848084727321701543312, −9.536892025614768893445639533023, −9.090568961835906240654095130431, −8.171388995562791192373802488120, −7.48541549843258920047192565774, −6.33122997704478630950608917603, −5.69727115956034935076832209417, −4.07214072444944193814729251169, −3.07016365568598367197415974624, −2.58818713210365071542160576364, 0.32614270682394806972251029428, 1.58526407203430459015873833937, 3.05366023045602494354423432102, 4.39481483342355958570986817301, 5.12145921713171550683598342011, 6.18551623522550468875229021520, 7.26929642963133938511034673000, 8.299694219515305493582904469230, 8.949416031443539489812368048587, 9.561676756081016220734361532054

Graph of the ZZ-function along the critical line