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Properties

Label 2-1216-19.7-c1-0-33
Degree 22
Conductor 12161216
Sign −0.980+0.194i-0.980 + 0.194i
Analytic cond. 9.709809.70980
Root an. cond. 3.116053.11605
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  + (−0.5 − 0.866i)3-s + (−2 − 3.46i)5-s + (1 − 1.73i)9-s + 3·11-s + (1 − 1.73i)13-s + (−1.99 + 3.46i)15-s + (−1 − 1.73i)17-s + (0.5 − 4.33i)19-s + (3 − 5.19i)23-s + (−5.49 + 9.52i)25-s − 5·27-s + (−2 + 3.46i)29-s + 10·31-s + (−1.5 − 2.59i)33-s − 2·37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.894 − 1.54i)5-s + (0.333 − 0.577i)9-s + 0.904·11-s + (0.277 − 0.480i)13-s + (−0.516 + 0.894i)15-s + (−0.242 − 0.420i)17-s + (0.114 − 0.993i)19-s + (0.625 − 1.08i)23-s + (−1.09 + 1.90i)25-s − 0.962·27-s + (−0.371 + 0.643i)29-s + 1.79·31-s + (−0.261 − 0.452i)33-s − 0.328·37-s + ⋯

Functional equation

Λ(s)=(1216s/2 ΓC(s) L(s)=((−0.980+0.194i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1216s/2 ΓC(s+1/2) L(s)=((−0.980+0.194i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12161216    =    26⋅192^{6} \cdot 19
Sign: −0.980+0.194i-0.980 + 0.194i
Analytic conductor: 9.709809.70980
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1216(577,⋅)\chi_{1216} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1216, ( :1/2), −0.980+0.194i)(2,\ 1216,\ (\ :1/2),\ -0.980 + 0.194i)

Particular Values

L(1)L(1) ≈\approx 1.1105863791.110586379
L(12)L(\frac12) ≈\approx 1.1105863791.110586379
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
19 1+(−0.5+4.33i)T 1 + (-0.5 + 4.33i)T
good3 1+(0.5+0.866i)T+(−1.5+2.59i)T2 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2}
5 1+(2+3.46i)T+(−2.5+4.33i)T2 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2}
7 1+7T2 1 + 7T^{2}
11 1−3T+11T2 1 - 3T + 11T^{2}
13 1+(−1+1.73i)T+(−6.5−11.2i)T2 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2}
17 1+(1+1.73i)T+(−8.5+14.7i)T2 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2}
23 1+(−3+5.19i)T+(−11.5−19.9i)T2 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2}
29 1+(2−3.46i)T+(−14.5−25.1i)T2 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2}
31 1−10T+31T2 1 - 10T + 31T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1+(4.5+7.79i)T+(−20.5+35.5i)T2 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2−3.46i)T+(−21.5+37.2i)T2 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2}
47 1+(6−10.3i)T+(−23.5−40.7i)T2 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2}
53 1+(1−1.73i)T+(−26.5−45.8i)T2 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2}
59 1+(−0.5−0.866i)T+(−29.5+51.0i)T2 1 + (-0.5 - 0.866i)T + (-29.5 + 51.0i)T^{2}
61 1+(4−6.92i)T+(−30.5−52.8i)T2 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2}
67 1+(4.5−7.79i)T+(−33.5−58.0i)T2 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2}
71 1+(3+5.19i)T+(−35.5+61.4i)T2 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2}
73 1+(−4.5−7.79i)T+(−36.5+63.2i)T2 1 + (-4.5 - 7.79i)T + (-36.5 + 63.2i)T^{2}
79 1+(2+3.46i)T+(−39.5+68.4i)T2 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2}
83 1+5T+83T2 1 + 5T + 83T^{2}
89 1+(−9+15.5i)T+(−44.5−77.0i)T2 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2}
97 1+(0.5+0.866i)T+(−48.5+84.0i)T2 1 + (0.5 + 0.866i)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

−8.948549508496821513849176904799, −8.788506542317774987935002109008, −7.70415298038578573845057256038, −6.91176462577580700595313155801, −6.06649326926350063125449392390, −4.84266368455744552217499572239, −4.36370824066764174070192684977, −3.19350881248401727017421363242, −1.33760290872410049724095802904, −0.55372918428696028243036519220, 1.84443473513052294899388421909, 3.31263948021534646629842438332, 3.88356393907147498616077250815, 4.83491829718995064412919922657, 6.18918864564920868474958882471, 6.73075461733201898288233715476, 7.64794508068527693644007941078, 8.287773417794300723168477318689, 9.588603199068747606250465893550, 10.19047343942992710068432091141

Graph of the ZZ-function along the critical line