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Properties

Label 2-325-65.3-c0-0-0
Degree 22
Conductor 325325
Sign 0.468−0.883i0.468 - 0.883i
Analytic cond. 0.1621960.162196
Root an. cond. 0.4027350.402735
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.258 + 0.965i)3-s + (−0.499 + 0.866i)6-s + (0.258 + 0.965i)7-s + (−0.707 − 0.707i)8-s + (−0.5 − 0.866i)11-s + (0.707 + 0.707i)13-s + i·14-s + (−0.5 − 0.866i)16-s + (−0.258 − 0.965i)17-s + (−0.866 − 0.5i)19-s − 21-s + (−0.258 − 0.965i)22-s + (0.258 − 0.965i)23-s + (0.866 − 0.499i)24-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.258 + 0.965i)3-s + (−0.499 + 0.866i)6-s + (0.258 + 0.965i)7-s + (−0.707 − 0.707i)8-s + (−0.5 − 0.866i)11-s + (0.707 + 0.707i)13-s + i·14-s + (−0.5 − 0.866i)16-s + (−0.258 − 0.965i)17-s + (−0.866 − 0.5i)19-s − 21-s + (−0.258 − 0.965i)22-s + (0.258 − 0.965i)23-s + (0.866 − 0.499i)24-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=((0.468−0.883i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s) L(s)=((0.468−0.883i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: 0.468−0.883i0.468 - 0.883i
Analytic conductor: 0.1621960.162196
Root analytic conductor: 0.4027350.402735
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ325(68,⋅)\chi_{325} (68, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 325, ( :0), 0.468−0.883i)(2,\ 325,\ (\ :0),\ 0.468 - 0.883i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0633287791.063328779
L(12)L(\frac12) ≈\approx 1.0633287791.063328779
L(1)L(1) 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 1
13 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good2 1+(−0.965−0.258i)T+(0.866+0.5i)T2 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2}
3 1+(0.258−0.965i)T+(−0.866−0.5i)T2 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2}
7 1+(−0.258−0.965i)T+(−0.866+0.5i)T2 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2}
11 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
17 1+(0.258+0.965i)T+(−0.866+0.5i)T2 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2}
19 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
23 1+(−0.258+0.965i)T+(−0.866−0.5i)T2 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2}
29 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
31 1+T2 1 + T^{2}
37 1+(0.965+0.258i)T+(0.866+0.5i)T2 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2}
41 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
43 1+(0.965−0.258i)T+(0.866−0.5i)T2 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2}
47 1−iT2 1 - iT^{2}
53 1+iT2 1 + iT^{2}
59 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
61 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.965+0.258i)T+(0.866+0.5i)T2 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2}
71 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
73 1+iT2 1 + iT^{2}
79 1−T2 1 - T^{2}
83 1+iT2 1 + iT^{2}
89 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
97 1+(−0.258−0.965i)T+(−0.866+0.5i)T2 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)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

−11.97745328251240419860787322072, −11.17716742363198603676242366172, −10.23426874134615930468138319292, −9.128359529591169715367184205955, −8.536078590989055215976771831270, −6.74392235497705062067962529038, −5.78941786480063852429820169715, −4.92200513007803894603375605824, −4.20004412422117194119859308834, −2.81826998948620157783634034250, 1.77247027103046769908399431599, 3.52621030662695976480044763924, 4.49912473471969393278016859786, 5.69785316935324921116678442293, 6.73378786824217088889563621048, 7.72360873780605816117183590160, 8.590562383179906133587738142728, 10.17083871556584762314021753885, 10.93301705732237245984978849436, 12.08997808522349553622929070982

Graph of the ZZ-function along the critical line