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Properties

Label 2-3332-3332.135-c0-0-1
Degree 22
Conductor 33323332
Sign 0.788−0.615i0.788 - 0.615i
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  + (−0.826 + 0.563i)2-s + (−0.997 + 0.925i)3-s + (0.365 − 0.930i)4-s + (0.302 − 1.32i)6-s + (0.974 − 0.222i)7-s + (0.222 + 0.974i)8-s + (0.0635 − 0.848i)9-s + (−0.0841 − 1.12i)11-s + (0.496 + 1.26i)12-s + (1.48 + 0.716i)13-s + (−0.680 + 0.733i)14-s + (−0.733 − 0.680i)16-s + (−0.988 − 0.149i)17-s + (0.425 + 0.736i)18-s + (−0.766 + 1.12i)21-s + (0.702 + 0.880i)22-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)2-s + (−0.997 + 0.925i)3-s + (0.365 − 0.930i)4-s + (0.302 − 1.32i)6-s + (0.974 − 0.222i)7-s + (0.222 + 0.974i)8-s + (0.0635 − 0.848i)9-s + (−0.0841 − 1.12i)11-s + (0.496 + 1.26i)12-s + (1.48 + 0.716i)13-s + (−0.680 + 0.733i)14-s + (−0.733 − 0.680i)16-s + (−0.988 − 0.149i)17-s + (0.425 + 0.736i)18-s + (−0.766 + 1.12i)21-s + (0.702 + 0.880i)22-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((0.788−0.615i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((0.788−0.615i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 0.788−0.615i0.788 - 0.615i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(135,⋅)\chi_{3332} (135, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.788−0.615i)(2,\ 3332,\ (\ :0),\ 0.788 - 0.615i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.68716676250.6871667625
L(12)L(\frac12) ≈\approx 0.68716676250.6871667625
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+(0.826−0.563i)T 1 + (0.826 - 0.563i)T
7 1+(−0.974+0.222i)T 1 + (-0.974 + 0.222i)T
17 1+(0.988+0.149i)T 1 + (0.988 + 0.149i)T
good3 1+(0.997−0.925i)T+(0.0747−0.997i)T2 1 + (0.997 - 0.925i)T + (0.0747 - 0.997i)T^{2}
5 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
11 1+(0.0841+1.12i)T+(−0.988+0.149i)T2 1 + (0.0841 + 1.12i)T + (-0.988 + 0.149i)T^{2}
13 1+(−1.48−0.716i)T+(0.623+0.781i)T2 1 + (-1.48 - 0.716i)T + (0.623 + 0.781i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(−0.858+0.129i)T+(0.955−0.294i)T2 1 + (-0.858 + 0.129i)T + (0.955 - 0.294i)T^{2}
29 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
31 1+(0.974+1.68i)T+(−0.5+0.866i)T2 1 + (0.974 + 1.68i)T + (-0.5 + 0.866i)T^{2}
37 1+(0.733−0.680i)T2 1 + (0.733 - 0.680i)T^{2}
41 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
43 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
47 1+(−0.365+0.930i)T2 1 + (-0.365 + 0.930i)T^{2}
53 1+(−0.698+1.77i)T+(−0.733−0.680i)T2 1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2}
59 1+(−0.826+0.563i)T2 1 + (-0.826 + 0.563i)T^{2}
61 1+(0.733−0.680i)T2 1 + (0.733 - 0.680i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1+(0.367+0.460i)T+(−0.222+0.974i)T2 1 + (0.367 + 0.460i)T + (-0.222 + 0.974i)T^{2}
73 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
79 1+(−0.997+1.72i)T+(−0.5−0.866i)T2 1 + (-0.997 + 1.72i)T + (-0.5 - 0.866i)T^{2}
83 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
89 1+(−0.109+1.46i)T+(−0.988−0.149i)T2 1 + (-0.109 + 1.46i)T + (-0.988 - 0.149i)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

−8.865147302321086401007807923064, −8.356687446892023947957248068065, −7.40878988605289469422696899339, −6.51279258841133665688157733622, −5.91868716172173508576143866859, −5.21810361843325993482173046195, −4.53834125789914560905395495492, −3.62659830537685171480210945874, −2.01107999415447734150929260852, −0.78403436064583527412418920883, 1.09895376787393161775256216249, 1.69633659702278923578541159927, 2.78718756952811722226016625516, 4.04412857388394656513096891107, 4.98538956853289925183770161165, 5.83346150391146038446882588573, 6.86117787210148242290715027570, 7.09489641885821884883580791834, 8.092596164706765649935201470398, 8.649059902684114477405934109052

Graph of the ZZ-function along the critical line