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Properties

Label 2-3332-3332.2447-c0-0-3
Degree 22
Conductor 33323332
Sign 0.481+0.876i0.481 + 0.876i
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  + (0.365 − 0.930i)2-s + (−0.0111 + 0.149i)3-s + (−0.733 − 0.680i)4-s + (0.134 + 0.0648i)6-s + (0.900 − 0.433i)7-s + (−0.900 + 0.433i)8-s + (0.966 + 0.145i)9-s + (0.722 − 0.108i)11-s + (0.109 − 0.101i)12-s + (0.455 + 0.571i)13-s + (−0.0747 − 0.997i)14-s + (0.0747 + 0.997i)16-s + (0.955 + 0.294i)17-s + (0.488 − 0.846i)18-s + (0.0546 + 0.139i)21-s + (0.162 − 0.712i)22-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)2-s + (−0.0111 + 0.149i)3-s + (−0.733 − 0.680i)4-s + (0.134 + 0.0648i)6-s + (0.900 − 0.433i)7-s + (−0.900 + 0.433i)8-s + (0.966 + 0.145i)9-s + (0.722 − 0.108i)11-s + (0.109 − 0.101i)12-s + (0.455 + 0.571i)13-s + (−0.0747 − 0.997i)14-s + (0.0747 + 0.997i)16-s + (0.955 + 0.294i)17-s + (0.488 − 0.846i)18-s + (0.0546 + 0.139i)21-s + (0.162 − 0.712i)22-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((0.481+0.876i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((0.481+0.876i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 0.481+0.876i0.481 + 0.876i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(2447,⋅)\chi_{3332} (2447, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.481+0.876i)(2,\ 3332,\ (\ :0),\ 0.481 + 0.876i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7570255681.757025568
L(12)L(\frac12) ≈\approx 1.7570255681.757025568
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+(−0.365+0.930i)T 1 + (-0.365 + 0.930i)T
7 1+(−0.900+0.433i)T 1 + (-0.900 + 0.433i)T
17 1+(−0.955−0.294i)T 1 + (-0.955 - 0.294i)T
good3 1+(0.0111−0.149i)T+(−0.988−0.149i)T2 1 + (0.0111 - 0.149i)T + (-0.988 - 0.149i)T^{2}
5 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
11 1+(−0.722+0.108i)T+(0.955−0.294i)T2 1 + (-0.722 + 0.108i)T + (0.955 - 0.294i)T^{2}
13 1+(−0.455−0.571i)T+(−0.222+0.974i)T2 1 + (-0.455 - 0.571i)T + (-0.222 + 0.974i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(1.19−0.367i)T+(0.826−0.563i)T2 1 + (1.19 - 0.367i)T + (0.826 - 0.563i)T^{2}
29 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
31 1+(0.900−1.56i)T+(−0.5−0.866i)T2 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2}
37 1+(−0.0747+0.997i)T2 1 + (-0.0747 + 0.997i)T^{2}
41 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
43 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
47 1+(0.733+0.680i)T2 1 + (0.733 + 0.680i)T^{2}
53 1+(1.21+1.12i)T+(0.0747+0.997i)T2 1 + (1.21 + 1.12i)T + (0.0747 + 0.997i)T^{2}
59 1+(−0.365+0.930i)T2 1 + (-0.365 + 0.930i)T^{2}
61 1+(−0.0747+0.997i)T2 1 + (-0.0747 + 0.997i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+(−0.367+1.61i)T+(−0.900−0.433i)T2 1 + (-0.367 + 1.61i)T + (-0.900 - 0.433i)T^{2}
73 1+(0.733−0.680i)T2 1 + (0.733 - 0.680i)T^{2}
79 1+(0.988+1.71i)T+(−0.5+0.866i)T2 1 + (0.988 + 1.71i)T + (-0.5 + 0.866i)T^{2}
83 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
89 1+(0.147+0.0222i)T+(0.955+0.294i)T2 1 + (0.147 + 0.0222i)T + (0.955 + 0.294i)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

−8.876518667982110483115303704246, −8.041821987511253315095597841089, −7.22133719107230246322451679055, −6.31933196898849435699427717901, −5.35571524392628963591512710234, −4.68437890968365946780229260621, −3.88316028410984038302803113404, −3.38298266180389679943794909545, −1.68784286700694512684485983916, −1.46237620828986753753954683686, 1.20428093689085394173066419885, 2.52562505145583689150304974782, 3.84472277537081628355889239674, 4.29804224163850949562555438930, 5.26510354460159993594255532510, 5.95653514296305028677058144967, 6.62030809972976821056019763691, 7.58739978104213017043647792654, 7.938446414148832100804882073013, 8.699648136904427596636333757796

Graph of the ZZ-function along the critical line