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Properties

Label 2-1134-7.4-c1-0-21
Degree 22
Conductor 11341134
Sign −0.574+0.818i-0.574 + 0.818i
Analytic cond. 9.055039.05503
Root an. cond. 3.009153.00915
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)2-s + (−0.499 + 0.866i)4-s + (0.230 + 0.398i)5-s + (−2.32 + 1.26i)7-s + 0.999·8-s + (0.230 − 0.398i)10-s + (1.82 − 3.15i)11-s − 1.46·13-s + (2.25 + 1.38i)14-s + (−0.5 − 0.866i)16-s + (−1.86 + 3.23i)17-s + (−2.02 − 3.51i)19-s − 0.460·20-s − 3.64·22-s + (−0.566 − 0.981i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.102 + 0.178i)5-s + (−0.878 + 0.478i)7-s + 0.353·8-s + (0.0728 − 0.126i)10-s + (0.549 − 0.952i)11-s − 0.405·13-s + (0.603 + 0.368i)14-s + (−0.125 − 0.216i)16-s + (−0.452 + 0.784i)17-s + (−0.465 − 0.805i)19-s − 0.102·20-s − 0.777·22-s + (−0.118 − 0.204i)23-s + ⋯

Functional equation

Λ(s)=(1134s/2 ΓC(s) L(s)=((−0.574+0.818i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1134s/2 ΓC(s+1/2) L(s)=((−0.574+0.818i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11341134    =    2⋅34⋅72 \cdot 3^{4} \cdot 7
Sign: −0.574+0.818i-0.574 + 0.818i
Analytic conductor: 9.055039.05503
Root analytic conductor: 3.009153.00915
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1134(487,⋅)\chi_{1134} (487, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1134, ( :1/2), −0.574+0.818i)(2,\ 1134,\ (\ :1/2),\ -0.574 + 0.818i)

Particular Values

L(1)L(1) ≈\approx 0.79909623990.7990962399
L(12)L(\frac12) ≈\approx 0.79909623990.7990962399
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1 1
7 1+(2.32−1.26i)T 1 + (2.32 - 1.26i)T
good5 1+(−0.230−0.398i)T+(−2.5+4.33i)T2 1 + (-0.230 - 0.398i)T + (-2.5 + 4.33i)T^{2}
11 1+(−1.82+3.15i)T+(−5.5−9.52i)T2 1 + (-1.82 + 3.15i)T + (-5.5 - 9.52i)T^{2}
13 1+1.46T+13T2 1 + 1.46T + 13T^{2}
17 1+(1.86−3.23i)T+(−8.5−14.7i)T2 1 + (1.86 - 3.23i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.02+3.51i)T+(−9.5+16.4i)T2 1 + (2.02 + 3.51i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.566+0.981i)T+(−11.5+19.9i)T2 1 + (0.566 + 0.981i)T + (-11.5 + 19.9i)T^{2}
29 1−8.97T+29T2 1 - 8.97T + 29T^{2}
31 1+(−0.257+0.445i)T+(−15.5−26.8i)T2 1 + (-0.257 + 0.445i)T + (-15.5 - 26.8i)T^{2}
37 1+(4.55+7.88i)T+(−18.5+32.0i)T2 1 + (4.55 + 7.88i)T + (-18.5 + 32.0i)T^{2}
41 1−0.945T+41T2 1 - 0.945T + 41T^{2}
43 1+9.32T+43T2 1 + 9.32T + 43T^{2}
47 1+(1.16+2.01i)T+(−23.5+40.7i)T2 1 + (1.16 + 2.01i)T + (-23.5 + 40.7i)T^{2}
53 1+(−6.21+10.7i)T+(−26.5−45.8i)T2 1 + (-6.21 + 10.7i)T + (-26.5 - 45.8i)T^{2}
59 1+(−6.44+11.1i)T+(−29.5−51.0i)T2 1 + (-6.44 + 11.1i)T + (-29.5 - 51.0i)T^{2}
61 1+(6.04+10.4i)T+(−30.5+52.8i)T2 1 + (6.04 + 10.4i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.16+2.00i)T+(−33.5−58.0i)T2 1 + (-1.16 + 2.00i)T + (-33.5 - 58.0i)T^{2}
71 1−1.67T+71T2 1 - 1.67T + 71T^{2}
73 1+(6.62−11.4i)T+(−36.5−63.2i)T2 1 + (6.62 - 11.4i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2.50−4.33i)T+(−39.5+68.4i)T2 1 + (-2.50 - 4.33i)T + (-39.5 + 68.4i)T^{2}
83 1+6.64T+83T2 1 + 6.64T + 83T^{2}
89 1+(1.36+2.36i)T+(−44.5+77.0i)T2 1 + (1.36 + 2.36i)T + (-44.5 + 77.0i)T^{2}
97 1−11.1T+97T2 1 - 11.1T + 97T^{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

−9.621004341105030715166194494771, −8.644240158644945047176214270723, −8.380054230551556400592415574403, −6.78439226911131595715010736153, −6.43196961011825598775704146582, −5.19519306690671494788902375234, −4.00584817743247724348132592096, −3.07625436060763517994838722589, −2.15340400706614926263789008071, −0.41508531589434567343333966809, 1.33172345444547014894403688882, 2.88533625539755533063736273890, 4.19448576483144773271991219079, 4.96299467513367235472222762136, 6.14085776457858723784441079590, 6.88601878420812743251630708524, 7.40378649262646684649841589186, 8.555801165934233218999919145966, 9.248191055705522846102719016229, 10.05604508814928196186744945862

Graph of the ZZ-function along the critical line