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Properties

Label 2-1260-105.2-c1-0-7
Degree 22
Conductor 12601260
Sign 0.259−0.965i0.259 - 0.965i
Analytic cond. 10.061110.0611
Root an. cond. 3.171933.17193
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  + (1.14 + 1.91i)5-s + (0.642 + 2.56i)7-s + (2.01 − 1.16i)11-s + (0.104 − 0.104i)13-s + (3.64 + 0.977i)17-s + (−0.646 − 0.373i)19-s + (−1.75 + 0.471i)23-s + (−2.37 + 4.40i)25-s + 7.38·29-s + (0.668 + 1.15i)31-s + (−4.19 + 4.17i)35-s + (−6.66 + 1.78i)37-s − 0.0508i·41-s + (−3.06 + 3.06i)43-s + (−0.746 − 2.78i)47-s + ⋯
L(s)  = 1  + (0.512 + 0.858i)5-s + (0.242 + 0.970i)7-s + (0.608 − 0.351i)11-s + (0.0289 − 0.0289i)13-s + (0.884 + 0.237i)17-s + (−0.148 − 0.0855i)19-s + (−0.366 + 0.0982i)23-s + (−0.474 + 0.880i)25-s + 1.37·29-s + (0.120 + 0.207i)31-s + (−0.708 + 0.705i)35-s + (−1.09 + 0.293i)37-s − 0.00794i·41-s + (−0.466 + 0.466i)43-s + (−0.108 − 0.406i)47-s + ⋯

Functional equation

Λ(s)=(1260s/2 ΓC(s) L(s)=((0.259−0.965i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1260s/2 ΓC(s+1/2) L(s)=((0.259−0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12601260    =    22⋅32⋅5⋅72^{2} \cdot 3^{2} \cdot 5 \cdot 7
Sign: 0.259−0.965i0.259 - 0.965i
Analytic conductor: 10.061110.0611
Root analytic conductor: 3.171933.17193
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1260(737,⋅)\chi_{1260} (737, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1260, ( :1/2), 0.259−0.965i)(2,\ 1260,\ (\ :1/2),\ 0.259 - 0.965i)

Particular Values

L(1)L(1) ≈\approx 1.8863615261.886361526
L(12)L(\frac12) ≈\approx 1.8863615261.886361526
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)
bad2 1 1
3 1 1
5 1+(−1.14−1.91i)T 1 + (-1.14 - 1.91i)T
7 1+(−0.642−2.56i)T 1 + (-0.642 - 2.56i)T
good11 1+(−2.01+1.16i)T+(5.5−9.52i)T2 1 + (-2.01 + 1.16i)T + (5.5 - 9.52i)T^{2}
13 1+(−0.104+0.104i)T−13iT2 1 + (-0.104 + 0.104i)T - 13iT^{2}
17 1+(−3.64−0.977i)T+(14.7+8.5i)T2 1 + (-3.64 - 0.977i)T + (14.7 + 8.5i)T^{2}
19 1+(0.646+0.373i)T+(9.5+16.4i)T2 1 + (0.646 + 0.373i)T + (9.5 + 16.4i)T^{2}
23 1+(1.75−0.471i)T+(19.9−11.5i)T2 1 + (1.75 - 0.471i)T + (19.9 - 11.5i)T^{2}
29 1−7.38T+29T2 1 - 7.38T + 29T^{2}
31 1+(−0.668−1.15i)T+(−15.5+26.8i)T2 1 + (-0.668 - 1.15i)T + (-15.5 + 26.8i)T^{2}
37 1+(6.66−1.78i)T+(32.0−18.5i)T2 1 + (6.66 - 1.78i)T + (32.0 - 18.5i)T^{2}
41 1+0.0508iT−41T2 1 + 0.0508iT - 41T^{2}
43 1+(3.06−3.06i)T−43iT2 1 + (3.06 - 3.06i)T - 43iT^{2}
47 1+(0.746+2.78i)T+(−40.7+23.5i)T2 1 + (0.746 + 2.78i)T + (-40.7 + 23.5i)T^{2}
53 1+(0.248−0.926i)T+(−45.8−26.5i)T2 1 + (0.248 - 0.926i)T + (-45.8 - 26.5i)T^{2}
59 1+(−3.37−5.83i)T+(−29.5+51.0i)T2 1 + (-3.37 - 5.83i)T + (-29.5 + 51.0i)T^{2}
61 1+(3.61−6.25i)T+(−30.5−52.8i)T2 1 + (3.61 - 6.25i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.62−9.80i)T+(−58.0−33.5i)T2 1 + (2.62 - 9.80i)T + (-58.0 - 33.5i)T^{2}
71 1+12.8iT−71T2 1 + 12.8iT - 71T^{2}
73 1+(−10.9−2.94i)T+(63.2+36.5i)T2 1 + (-10.9 - 2.94i)T + (63.2 + 36.5i)T^{2}
79 1+(9.97+5.75i)T+(39.5+68.4i)T2 1 + (9.97 + 5.75i)T + (39.5 + 68.4i)T^{2}
83 1+(0.461+0.461i)T+83iT2 1 + (0.461 + 0.461i)T + 83iT^{2}
89 1+(−2.21+3.83i)T+(−44.5−77.0i)T2 1 + (-2.21 + 3.83i)T + (-44.5 - 77.0i)T^{2}
97 1+(−5.18−5.18i)T+97iT2 1 + (-5.18 - 5.18i)T + 97iT^{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

−9.984573212596237024866985147136, −8.992477465747398745947162674336, −8.367111180918252984136189018753, −7.34489065153569594582457171220, −6.38606455775029316604528229701, −5.85217748354744344150888441770, −4.88723599905737613048938713218, −3.53660993396302169448812062175, −2.69999420199489241660730104717, −1.55269245892348981710132955738, 0.861927074764382066671901683378, 1.91127401346962972640462465394, 3.47905732147217570244527732257, 4.44678486069065553881318326953, 5.15095665086919394396961522763, 6.21847741216002102223053605035, 7.03721635749029601538058820026, 7.996454129472163803076212602328, 8.661445470827627029448329019153, 9.690184497861766841969599449947

Graph of the ZZ-function along the critical line