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Properties

Label 2-1232-1232.1021-c0-0-1
Degree 22
Conductor 12321232
Sign 0.875+0.482i0.875 + 0.482i
Analytic cond. 0.6148480.614848
Root an. cond. 0.7841220.784122
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  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (0.951 + 0.309i)9-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + 0.999·18-s + (−0.587 − 0.809i)22-s − 1.17i·23-s + (0.587 + 0.809i)25-s + 0.999i·28-s + (0.278 + 1.76i)29-s − i·32-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (0.951 + 0.309i)9-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + 0.999·18-s + (−0.587 − 0.809i)22-s − 1.17i·23-s + (0.587 + 0.809i)25-s + 0.999i·28-s + (0.278 + 1.76i)29-s − i·32-s + ⋯

Functional equation

Λ(s)=(1232s/2 ΓC(s) L(s)=((0.875+0.482i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1232s/2 ΓC(s) L(s)=((0.875+0.482i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12321232    =    24⋅7⋅112^{4} \cdot 7 \cdot 11
Sign: 0.875+0.482i0.875 + 0.482i
Analytic conductor: 0.6148480.614848
Root analytic conductor: 0.7841220.784122
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1232(1021,⋅)\chi_{1232} (1021, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1232, ( :0), 0.875+0.482i)(2,\ 1232,\ (\ :0),\ 0.875 + 0.482i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8653188661.865318866
L(12)L(\frac12) ≈\approx 1.8653188661.865318866
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+(−0.951+0.309i)T 1 + (-0.951 + 0.309i)T
7 1+(0.587−0.809i)T 1 + (0.587 - 0.809i)T
11 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
good3 1+(−0.951−0.309i)T2 1 + (-0.951 - 0.309i)T^{2}
5 1+(−0.587−0.809i)T2 1 + (-0.587 - 0.809i)T^{2}
13 1+(−0.587+0.809i)T2 1 + (-0.587 + 0.809i)T^{2}
17 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
19 1+(−0.951−0.309i)T2 1 + (-0.951 - 0.309i)T^{2}
23 1+1.17iT−T2 1 + 1.17iT - T^{2}
29 1+(−0.278−1.76i)T+(−0.951+0.309i)T2 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)T^{2}
31 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
37 1+(1.95−0.309i)T+(0.951−0.309i)T2 1 + (1.95 - 0.309i)T + (0.951 - 0.309i)T^{2}
41 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
43 1+(0.642−0.642i)T−iT2 1 + (0.642 - 0.642i)T - iT^{2}
47 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
53 1+(0.809+1.58i)T+(−0.587+0.809i)T2 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2}
59 1+(−0.951+0.309i)T2 1 + (-0.951 + 0.309i)T^{2}
61 1+(0.587+0.809i)T2 1 + (0.587 + 0.809i)T^{2}
67 1+(0.642+0.642i)T+iT2 1 + (0.642 + 0.642i)T + iT^{2}
71 1+(0.587−0.190i)T+(0.809−0.587i)T2 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2}
73 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
79 1+(0.587−1.80i)T+(−0.809−0.587i)T2 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2}
83 1+(0.587+0.809i)T2 1 + (0.587 + 0.809i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)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

−10.13099298717208022045046884265, −9.100622778709531610999326435724, −8.260868927870746963481120896596, −6.98312431453464783214564159385, −6.52378768227714903527470193663, −5.40546674427729166205077782113, −4.86712262124318436102262148960, −3.56651769700459456030085291234, −2.87605623009912368372891417060, −1.60230024872754652635835282050, 1.75611397364996184816687780645, 3.10451404133011712318006348524, 4.09001873629358783445089346108, 4.65154950449318336660727364156, 5.80703529786564639735295238317, 6.76790173811938826154264074259, 7.24626156038322532847174475142, 7.974498391269328844655176678483, 9.282446574469930032837150449049, 10.18640728123634574609155687695

Graph of the ZZ-function along the critical line