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Properties

Label 2-3332-3332.407-c0-0-3
Degree 22
Conductor 33323332
Sign −0.801−0.598i-0.801 - 0.598i
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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.222 − 0.974i)2-s + (0.974 − 1.22i)3-s + (−0.900 − 0.433i)4-s + (−0.974 − 1.22i)6-s + (−0.781 − 0.623i)7-s + (−0.623 + 0.781i)8-s + (−0.321 − 1.40i)9-s + (0.433 − 1.90i)11-s + (−1.40 + 0.678i)12-s + (−0.0990 + 0.433i)13-s + (−0.781 + 0.623i)14-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s − 1.44·18-s + (−1.52 + 0.347i)21-s + (−1.75 − 0.846i)22-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (0.974 − 1.22i)3-s + (−0.900 − 0.433i)4-s + (−0.974 − 1.22i)6-s + (−0.781 − 0.623i)7-s + (−0.623 + 0.781i)8-s + (−0.321 − 1.40i)9-s + (0.433 − 1.90i)11-s + (−1.40 + 0.678i)12-s + (−0.0990 + 0.433i)13-s + (−0.781 + 0.623i)14-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s − 1.44·18-s + (−1.52 + 0.347i)21-s + (−1.75 − 0.846i)22-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((−0.801−0.598i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((−0.801−0.598i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: −0.801−0.598i-0.801 - 0.598i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(407,⋅)\chi_{3332} (407, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), −0.801−0.598i)(2,\ 3332,\ (\ :0),\ -0.801 - 0.598i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4419684621.441968462
L(12)L(\frac12) ≈\approx 1.4419684621.441968462
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+(−0.222+0.974i)T 1 + (-0.222 + 0.974i)T
7 1+(0.781+0.623i)T 1 + (0.781 + 0.623i)T
17 1+(0.900−0.433i)T 1 + (0.900 - 0.433i)T
good3 1+(−0.974+1.22i)T+(−0.222−0.974i)T2 1 + (-0.974 + 1.22i)T + (-0.222 - 0.974i)T^{2}
5 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
11 1+(−0.433+1.90i)T+(−0.900−0.433i)T2 1 + (-0.433 + 1.90i)T + (-0.900 - 0.433i)T^{2}
13 1+(0.0990−0.433i)T+(−0.900−0.433i)T2 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2}
19 1−T2 1 - T^{2}
23 1+(−1.75−0.846i)T+(0.623+0.781i)T2 1 + (-1.75 - 0.846i)T + (0.623 + 0.781i)T^{2}
29 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
31 1+1.56T+T2 1 + 1.56T + T^{2}
37 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
41 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
43 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
47 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
53 1+(1.12+0.541i)T+(0.623+0.781i)T2 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2}
59 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
61 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
67 1−T2 1 - T^{2}
71 1+(−1.40−0.678i)T+(0.623+0.781i)T2 1 + (-1.40 - 0.678i)T + (0.623 + 0.781i)T^{2}
73 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
79 1−1.94T+T2 1 - 1.94T + T^{2}
83 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
89 1+(−0.277−1.21i)T+(−0.900+0.433i)T2 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)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

−8.524691492170903427233193785782, −7.79085357911255892002101448776, −6.75910726388546770122495843532, −6.33568483139966082694483107763, −5.29932559112299617042797391408, −3.95971669711100366207224469614, −3.40085986949927219624964267328, −2.73343573491181054499224866888, −1.68636441992371162168697999859, −0.71123287276723704805990679413, 2.27245219337206432788446173304, 3.19682426352975954174877618798, 3.89270618416553483477381123303, 4.84200406741456707675548547400, 5.09265598927836257715264861051, 6.38458859493957620044091823697, 7.07155081868826347285518930436, 7.70264140682374544988197389849, 8.809816852457762940966524483293, 9.207184294612230344918471480224

Graph of the ZZ-function along the critical line