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Properties

Label 2-2523-87.53-c0-0-2
Degree 22
Conductor 25232523
Sign −0.799−0.600i-0.799 - 0.600i
Analytic cond. 1.259141.25914
Root an. cond. 1.122111.12211
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  + (0.222 + 0.974i)3-s + (−0.222 + 0.974i)4-s + (−0.137 − 0.602i)7-s + (−0.900 + 0.433i)9-s − 12-s + (1.45 + 0.702i)13-s + (−0.900 − 0.433i)16-s + (−0.360 + 1.57i)19-s + (0.556 − 0.268i)21-s + (−0.222 + 0.974i)25-s + (−0.623 − 0.781i)27-s + 0.618·28-s + (−0.385 − 0.483i)31-s + (−0.222 − 0.974i)36-s + (−1.45 + 0.702i)37-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)3-s + (−0.222 + 0.974i)4-s + (−0.137 − 0.602i)7-s + (−0.900 + 0.433i)9-s − 12-s + (1.45 + 0.702i)13-s + (−0.900 − 0.433i)16-s + (−0.360 + 1.57i)19-s + (0.556 − 0.268i)21-s + (−0.222 + 0.974i)25-s + (−0.623 − 0.781i)27-s + 0.618·28-s + (−0.385 − 0.483i)31-s + (−0.222 − 0.974i)36-s + (−1.45 + 0.702i)37-s + ⋯

Functional equation

Λ(s)=(2523s/2 ΓC(s) L(s)=((−0.799−0.600i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2523s/2 ΓC(s) L(s)=((−0.799−0.600i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25232523    =    3⋅2923 \cdot 29^{2}
Sign: −0.799−0.600i-0.799 - 0.600i
Analytic conductor: 1.259141.25914
Root analytic conductor: 1.122111.12211
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2523(1619,⋅)\chi_{2523} (1619, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2523, ( :0), −0.799−0.600i)(2,\ 2523,\ (\ :0),\ -0.799 - 0.600i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0620198071.062019807
L(12)L(\frac12) ≈\approx 1.0620198071.062019807
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)
bad3 1+(−0.222−0.974i)T 1 + (-0.222 - 0.974i)T
29 1 1
good2 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
5 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
7 1+(0.137+0.602i)T+(−0.900+0.433i)T2 1 + (0.137 + 0.602i)T + (-0.900 + 0.433i)T^{2}
11 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
13 1+(−1.45−0.702i)T+(0.623+0.781i)T2 1 + (-1.45 - 0.702i)T + (0.623 + 0.781i)T^{2}
17 1−T2 1 - T^{2}
19 1+(0.360−1.57i)T+(−0.900−0.433i)T2 1 + (0.360 - 1.57i)T + (-0.900 - 0.433i)T^{2}
23 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
31 1+(0.385+0.483i)T+(−0.222+0.974i)T2 1 + (0.385 + 0.483i)T + (-0.222 + 0.974i)T^{2}
37 1+(1.45−0.702i)T+(0.623−0.781i)T2 1 + (1.45 - 0.702i)T + (0.623 - 0.781i)T^{2}
41 1−T2 1 - T^{2}
43 1+(0.385−0.483i)T+(−0.222−0.974i)T2 1 + (0.385 - 0.483i)T + (-0.222 - 0.974i)T^{2}
47 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
53 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
59 1−T2 1 - T^{2}
61 1+(−0.137−0.602i)T+(−0.900+0.433i)T2 1 + (-0.137 - 0.602i)T + (-0.900 + 0.433i)T^{2}
67 1+(−1.45+0.702i)T+(0.623−0.781i)T2 1 + (-1.45 + 0.702i)T + (0.623 - 0.781i)T^{2}
71 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
73 1+(−1.00+1.26i)T+(−0.222−0.974i)T2 1 + (-1.00 + 1.26i)T + (-0.222 - 0.974i)T^{2}
79 1+(1.45−0.702i)T+(0.623−0.781i)T2 1 + (1.45 - 0.702i)T + (0.623 - 0.781i)T^{2}
83 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
89 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
97 1+(−0.137+0.602i)T+(−0.900−0.433i)T2 1 + (-0.137 + 0.602i)T + (-0.900 - 0.433i)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

−9.330434680158206463753683081357, −8.565394140714248515088762549070, −8.153448204026573665598849787855, −7.23038715329502711906527708325, −6.30707818073237905693459084773, −5.38903018215547375727880013399, −4.31577758418334292058710297518, −3.73670710440203855398306936498, −3.30109526937671002171357154317, −1.82483046788889447936896001604, 0.68919741301323215836102271620, 1.85718642364433891097444640928, 2.76112606368006930134856483156, 3.88298804307348861220733058308, 5.17692532281506499739022334287, 5.73864635660222897462014068625, 6.50583995274447043944455641313, 7.03478464279088671034613109677, 8.277959037060116206959108451852, 8.719269721603570173781310991950

Graph of the ZZ-function along the critical line