Properties

Label 2-45e2-225.146-c0-0-0
Degree 22
Conductor 20252025
Sign 0.4320.901i0.432 - 0.901i
Analytic cond. 1.010601.01060
Root an. cond. 1.005281.00528
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  + (−0.459 + 0.413i)2-s + (−0.0646 + 0.614i)4-s + (−0.743 − 0.669i)5-s + (−0.309 + 0.535i)7-s + (−0.587 − 0.809i)8-s + 0.618·10-s + (0.743 − 0.669i)11-s + (−0.0794 − 0.373i)14-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (0.459 − 0.413i)20-s + (−0.0646 + 0.614i)22-s + (0.207 + 0.978i)23-s + (0.104 + 0.994i)25-s + (−0.309 − 0.224i)28-s + (−0.658 − 1.47i)29-s + ⋯
L(s)  = 1  + (−0.459 + 0.413i)2-s + (−0.0646 + 0.614i)4-s + (−0.743 − 0.669i)5-s + (−0.309 + 0.535i)7-s + (−0.587 − 0.809i)8-s + 0.618·10-s + (0.743 − 0.669i)11-s + (−0.0794 − 0.373i)14-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (0.459 − 0.413i)20-s + (−0.0646 + 0.614i)22-s + (0.207 + 0.978i)23-s + (0.104 + 0.994i)25-s + (−0.309 − 0.224i)28-s + (−0.658 − 1.47i)29-s + ⋯

Functional equation

Λ(s)=(2025s/2ΓC(s)L(s)=((0.4320.901i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2025s/2ΓC(s)L(s)=((0.4320.901i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20252025    =    34523^{4} \cdot 5^{2}
Sign: 0.4320.901i0.432 - 0.901i
Analytic conductor: 1.010601.01060
Root analytic conductor: 1.005281.00528
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2025(1646,)\chi_{2025} (1646, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2025, ( :0), 0.4320.901i)(2,\ 2025,\ (\ :0),\ 0.432 - 0.901i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.74782137150.7478213715
L(12)L(\frac12) \approx 0.74782137150.7478213715
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1+(0.743+0.669i)T 1 + (0.743 + 0.669i)T
good2 1+(0.4590.413i)T+(0.1040.994i)T2 1 + (0.459 - 0.413i)T + (0.104 - 0.994i)T^{2}
7 1+(0.3090.535i)T+(0.50.866i)T2 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2}
11 1+(0.743+0.669i)T+(0.1040.994i)T2 1 + (-0.743 + 0.669i)T + (0.104 - 0.994i)T^{2}
13 1+(0.1040.994i)T2 1 + (-0.104 - 0.994i)T^{2}
17 1+(0.5870.809i)T+(0.309+0.951i)T2 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2}
19 1+(0.809+0.587i)T+(0.3090.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2}
23 1+(0.2070.978i)T+(0.913+0.406i)T2 1 + (-0.207 - 0.978i)T + (-0.913 + 0.406i)T^{2}
29 1+(0.658+1.47i)T+(0.669+0.743i)T2 1 + (0.658 + 1.47i)T + (-0.669 + 0.743i)T^{2}
31 1+(0.669+0.743i)T2 1 + (0.669 + 0.743i)T^{2}
37 1+(0.51.53i)T+(0.809+0.587i)T2 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2}
41 1+(0.7430.669i)T+(0.104+0.994i)T2 1 + (-0.743 - 0.669i)T + (0.104 + 0.994i)T^{2}
43 1+(0.50.866i)T2 1 + (-0.5 - 0.866i)T^{2}
47 1+(0.6581.47i)T+(0.669+0.743i)T2 1 + (-0.658 - 1.47i)T + (-0.669 + 0.743i)T^{2}
53 1+(0.363+0.5i)T+(0.3090.951i)T2 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2}
59 1+(0.104+0.994i)T2 1 + (0.104 + 0.994i)T^{2}
61 1+(0.669+0.743i)T+(0.104+0.994i)T2 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2}
67 1+(0.5640.251i)T+(0.669+0.743i)T2 1 + (-0.564 - 0.251i)T + (0.669 + 0.743i)T^{2}
71 1+(0.9511.30i)T+(0.3090.951i)T2 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2}
73 1+(0.8090.587i)T2 1 + (-0.809 - 0.587i)T^{2}
79 1+(1.47+0.658i)T+(0.6690.743i)T2 1 + (-1.47 + 0.658i)T + (0.669 - 0.743i)T^{2}
83 1+(0.6140.0646i)T+(0.9780.207i)T2 1 + (0.614 - 0.0646i)T + (0.978 - 0.207i)T^{2}
89 1+(0.951+0.309i)T+(0.809+0.587i)T2 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2}
97 1+(0.913+0.406i)T+(0.6690.743i)T2 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.309244999148642232511590786198, −8.640408559550704426187333559223, −7.916721242263277598640020760354, −7.45592589382914959748508356035, −6.35029170978153427157918742635, −5.70154971532837151137747251170, −4.47770103010308932400639739405, −3.67245820294498941245492625176, −2.92867718632348906129502072146, −1.08400043211464697062822940920, 0.816459609364607724552658043149, 2.17808236767767896473866573445, 3.29215037107903319855970567139, 4.12540859854794156419175221802, 5.17789684325368895422988463833, 6.09412751138016942186837549920, 7.15489814993628706706910706983, 7.37993170183307506425902439429, 8.639428462234207713911909929002, 9.310114014570906467371382051734

Graph of the ZZ-function along the critical line