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Properties

Label 2-1596-1596.83-c0-0-2
Degree 22
Conductor 15961596
Sign 0.977+0.211i0.977 + 0.211i
Analytic cond. 0.7965070.796507
Root an. cond. 0.8924720.892472
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  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.499 − 0.866i)6-s + 7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)10-s + 11-s + 0.999·12-s + (0.5 + 0.866i)14-s + (−0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s − 0.999·18-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.499 − 0.866i)6-s + 7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)10-s + 11-s + 0.999·12-s + (0.5 + 0.866i)14-s + (−0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s − 0.999·18-s + ⋯

Functional equation

Λ(s)=(1596s/2 ΓC(s) L(s)=((0.977+0.211i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1596s/2 ΓC(s) L(s)=((0.977+0.211i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15961596    =    22⋅3⋅7⋅192^{2} \cdot 3 \cdot 7 \cdot 19
Sign: 0.977+0.211i0.977 + 0.211i
Analytic conductor: 0.7965070.796507
Root analytic conductor: 0.8924720.892472
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1596(83,⋅)\chi_{1596} (83, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1596, ( :0), 0.977+0.211i)(2,\ 1596,\ (\ :0),\ 0.977 + 0.211i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1567058451.156705845
L(12)L(\frac12) ≈\approx 1.1567058451.156705845
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
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
7 1−T 1 - T
19 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good5 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
11 1−T+T2 1 - T + T^{2}
13 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
29 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
31 1−2T+T2 1 - 2T + T^{2}
37 1+T+T2 1 + T + T^{2}
41 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
43 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (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+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)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+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
97 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)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

−9.060283980223719389266635941910, −8.383838797923700874884287492424, −8.103225046207645987830121605288, −6.91170414691845306254359946523, −6.58000993713222950167333036663, −5.40792953057127450588370751030, −4.74280336816020163240315817064, −4.13596127480650960361179811252, −2.50997736454167739372411746163, −0.972733321512063517460296557456, 1.44260092666158496693050292945, 2.92932280342745345758789338625, 3.74399984198466322174026167947, 4.50047523684362175703236578705, 5.19377351736445941863355114817, 6.27053216134840864865462649175, 6.94300837519052736787524490748, 8.329647868614934950875684090772, 9.067886951137611438718453957655, 9.841925487486072338712976126604

Graph of the ZZ-function along the critical line