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Properties

Label 2-3332-476.135-c0-0-14
Degree 22
Conductor 33323332
Sign −0.991−0.126i-0.991 - 0.126i
Analytic cond. 1.662881.66288
Root an. cond. 1.289521.28952
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + (0.5 − 0.866i)9-s − 2·13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.5 − 0.866i)25-s + (−1 + 1.73i)26-s + (0.499 + 0.866i)32-s − 0.999·34-s − 0.999·36-s − 0.999·50-s + (0.999 + 1.73i)52-s + (−1 − 1.73i)53-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + (0.5 − 0.866i)9-s − 2·13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.5 − 0.866i)25-s + (−1 + 1.73i)26-s + (0.499 + 0.866i)32-s − 0.999·34-s − 0.999·36-s − 0.999·50-s + (0.999 + 1.73i)52-s + (−1 − 1.73i)53-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((−0.991−0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((−0.991−0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: −0.991−0.126i-0.991 - 0.126i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(2039,⋅)\chi_{3332} (2039, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), −0.991−0.126i)(2,\ 3332,\ (\ :0),\ -0.991 - 0.126i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.94866399210.9486639921
L(12)L(\frac12) ≈\approx 0.94866399210.9486639921
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)
bad2 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1 1
17 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good3 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
13 1+2T+T2 1 + 2T + T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)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+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1+T2 1 + T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
97 1−T2 1 - 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

−8.666240368694767713621973908942, −7.57318488192877308073674614286, −6.83823250939125812839740791210, −6.10221722846231853893072596094, −5.00527919875885398536363600171, −4.61226772355878742124274408018, −3.65039299881290295411485060490, −2.72259065586524609983632754041, −1.95152508359678581101723423990, −0.44385434784124211530261702712, 1.97053740060680400843865440243, 2.89528181419206752831906828286, 4.05679493636604253725648459282, 4.72767493214349438419493053847, 5.31762033894491866688723147196, 6.19367765564667667915382483722, 7.10413381048564031751491802937, 7.55801255485593888238791729295, 8.147287188195646889761782045648, 9.118221012211938747699426161632

Graph of the ZZ-function along the critical line