Properties

Label 2-2695-385.109-c0-0-2
Degree 22
Conductor 26952695
Sign 0.328+0.944i0.328 + 0.944i
Analytic cond. 1.344981.34498
Root an. cond. 1.159731.15973
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  + (−1.22 − 0.707i)3-s + (0.5 − 0.866i)4-s + (−0.258 + 0.965i)5-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (−1.22 + 0.707i)12-s + (1 − 0.999i)15-s + (−0.499 − 0.866i)16-s + (0.707 + 0.707i)20-s + (1.73 − i)23-s + (−0.866 − 0.499i)25-s + (0.707 − 1.22i)31-s + (1.22 − 0.707i)33-s + 0.999·36-s + (0.499 + 0.866i)44-s + (−0.965 + 0.258i)45-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)3-s + (0.5 − 0.866i)4-s + (−0.258 + 0.965i)5-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (−1.22 + 0.707i)12-s + (1 − 0.999i)15-s + (−0.499 − 0.866i)16-s + (0.707 + 0.707i)20-s + (1.73 − i)23-s + (−0.866 − 0.499i)25-s + (0.707 − 1.22i)31-s + (1.22 − 0.707i)33-s + 0.999·36-s + (0.499 + 0.866i)44-s + (−0.965 + 0.258i)45-s + ⋯

Functional equation

Λ(s)=(2695s/2ΓC(s)L(s)=((0.328+0.944i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2695s/2ΓC(s)L(s)=((0.328+0.944i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26952695    =    572115 \cdot 7^{2} \cdot 11
Sign: 0.328+0.944i0.328 + 0.944i
Analytic conductor: 1.344981.34498
Root analytic conductor: 1.159731.15973
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2695(2419,)\chi_{2695} (2419, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2695, ( :0), 0.328+0.944i)(2,\ 2695,\ (\ :0),\ 0.328 + 0.944i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.78299967840.7829996784
L(12)L(\frac12) \approx 0.78299967840.7829996784
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(0.2580.965i)T 1 + (0.258 - 0.965i)T
7 1 1
11 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
good2 1+(0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
3 1+(1.22+0.707i)T+(0.5+0.866i)T2 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2}
13 1+T2 1 + T^{2}
17 1+(0.50.866i)T2 1 + (-0.5 - 0.866i)T^{2}
19 1+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(1.73+i)T+(0.50.866i)T2 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2}
29 1T2 1 - T^{2}
31 1+(0.707+1.22i)T+(0.50.866i)T2 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2}
37 1+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
41 1T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1+(1.22+0.707i)T+(0.50.866i)T2 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}
53 1+(1.73i)T+(0.5+0.866i)T2 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2}
59 1+(0.707+1.22i)T+(0.50.866i)T2 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2}
61 1+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(1.73+i)T+(0.5+0.866i)T2 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2}
71 1+T2 1 + T^{2}
73 1+(0.50.866i)T2 1 + (-0.5 - 0.866i)T^{2}
79 1+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(0.707+1.22i)T+(0.5+0.866i)T2 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2}
97 11.41iTT2 1 - 1.41iT - T^{2}
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   L(s)=p j=12(1αj,pps)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

−8.952565841678309393584710467693, −7.64076060784873612259706004692, −7.12782232833497550880412687891, −6.61084890592465899042443091408, −5.95070040028989505877822276498, −5.22411615682836248400278104398, −4.38262605739193287721777198036, −2.84830007165578642747408836879, −2.05244296524252265900292960449, −0.71885221688529286051645531665, 1.07305421619055593239645515158, 2.77749679417656683171763146478, 3.72067288005102341631711277355, 4.54788257366943927555891010175, 5.31577927695807387920028645600, 5.85081032061875640082497510171, 6.90661779813281065210807759010, 7.60848900579968908989184195382, 8.592510303444214183176486170027, 8.947484832249715924605580129400

Graph of the ZZ-function along the critical line