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Properties

Label 2-35e2-35.23-c0-0-2
Degree 22
Conductor 12251225
Sign 0.813+0.581i0.813 + 0.581i
Analytic cond. 0.6113540.611354
Root an. cond. 0.7818910.781891
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  + (0.866 − 0.5i)4-s + (0.866 + 0.5i)9-s + (−1 − 1.73i)11-s + (0.499 − 0.866i)16-s + 2i·29-s + 0.999·36-s + (−1.73 − 0.999i)44-s − 0.999i·64-s − 2·71-s + (1.73 + i)79-s + (0.499 + 0.866i)81-s − 1.99i·99-s + (−1.73 + i)109-s + (1 + 1.73i)116-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)4-s + (0.866 + 0.5i)9-s + (−1 − 1.73i)11-s + (0.499 − 0.866i)16-s + 2i·29-s + 0.999·36-s + (−1.73 − 0.999i)44-s − 0.999i·64-s − 2·71-s + (1.73 + i)79-s + (0.499 + 0.866i)81-s − 1.99i·99-s + (−1.73 + i)109-s + (1 + 1.73i)116-s + ⋯

Functional equation

Λ(s)=(1225s/2 ΓC(s) L(s)=((0.813+0.581i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1225s/2 ΓC(s) L(s)=((0.813+0.581i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12251225    =    52⋅725^{2} \cdot 7^{2}
Sign: 0.813+0.581i0.813 + 0.581i
Analytic conductor: 0.6113540.611354
Root analytic conductor: 0.7818910.781891
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1225(618,⋅)\chi_{1225} (618, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1225, ( :0), 0.813+0.581i)(2,\ 1225,\ (\ :0),\ 0.813 + 0.581i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2975086951.297508695
L(12)L(\frac12) ≈\approx 1.2975086951.297508695
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
7 1 1
good2 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
3 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
11 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
13 1+iT2 1 + iT^{2}
17 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
29 1−2iT−T2 1 - 2iT - T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
41 1+T2 1 + T^{2}
43 1+iT2 1 + iT^{2}
47 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
53 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
67 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
71 1+2T+T2 1 + 2T + T^{2}
73 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
79 1+(−1.73−i)T+(0.5+0.866i)T2 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2}
83 1+iT2 1 + iT^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1−iT2 1 - iT^{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

−10.10842161674811469590070569971, −9.010575697299198735540361607344, −8.102313856268269707268245424081, −7.36073443779792664314877648857, −6.52911788623658706797619758861, −5.61815753668855137836644432381, −4.97329414844732307864484567044, −3.49584534378517935283434112741, −2.58310175237063391028245005097, −1.29453476774021505052523383517, 1.80357785514661503823830905434, 2.66488087310503864355433301927, 3.94965068992227505423442553231, 4.73532175608862269840415202864, 5.98991009960892986026453138808, 6.88038970613669655230438498838, 7.50046853287663568076180233908, 8.070986283820880948501133195982, 9.373894503014709617359903289575, 10.06624578909594598452719613275

Graph of the ZZ-function along the critical line