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Properties

Label 2-2523-87.35-c0-0-3
Degree 22
Conductor 25232523
Sign 0.768+0.639i0.768 + 0.639i
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  + (−0.974 − 0.222i)3-s + (0.222 + 0.974i)4-s + (0.360 − 1.57i)7-s + (0.900 + 0.433i)9-s − i·12-s + (0.556 − 0.268i)13-s + (−0.900 + 0.433i)16-s + (0.602 − 0.137i)19-s + (−0.702 + 1.45i)21-s + (−0.222 − 0.974i)25-s + (−0.781 − 0.623i)27-s + 1.61·28-s + (−1.26 − 1.00i)31-s + (−0.222 + 0.974i)36-s + (−0.268 + 0.556i)37-s + ⋯
L(s)  = 1  + (−0.974 − 0.222i)3-s + (0.222 + 0.974i)4-s + (0.360 − 1.57i)7-s + (0.900 + 0.433i)9-s − i·12-s + (0.556 − 0.268i)13-s + (−0.900 + 0.433i)16-s + (0.602 − 0.137i)19-s + (−0.702 + 1.45i)21-s + (−0.222 − 0.974i)25-s + (−0.781 − 0.623i)27-s + 1.61·28-s + (−1.26 − 1.00i)31-s + (−0.222 + 0.974i)36-s + (−0.268 + 0.556i)37-s + ⋯

Functional equation

Λ(s)=(2523s/2 ΓC(s) L(s)=((0.768+0.639i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2523s/2 ΓC(s) L(s)=((0.768+0.639i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25232523    =    3⋅2923 \cdot 29^{2}
Sign: 0.768+0.639i0.768 + 0.639i
Analytic conductor: 1.259141.25914
Root analytic conductor: 1.122111.12211
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2523(1949,⋅)\chi_{2523} (1949, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2523, ( :0), 0.768+0.639i)(2,\ 2523,\ (\ :0),\ 0.768 + 0.639i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.94790891100.9479089110
L(12)L(\frac12) ≈\approx 0.94790891100.9479089110
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+(0.974+0.222i)T 1 + (0.974 + 0.222i)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+(−0.360+1.57i)T+(−0.900−0.433i)T2 1 + (-0.360 + 1.57i)T + (-0.900 - 0.433i)T^{2}
11 1+(0.623−0.781i)T2 1 + (0.623 - 0.781i)T^{2}
13 1+(−0.556+0.268i)T+(0.623−0.781i)T2 1 + (-0.556 + 0.268i)T + (0.623 - 0.781i)T^{2}
17 1+T2 1 + T^{2}
19 1+(−0.602+0.137i)T+(0.900−0.433i)T2 1 + (-0.602 + 0.137i)T + (0.900 - 0.433i)T^{2}
23 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
31 1+(1.26+1.00i)T+(0.222+0.974i)T2 1 + (1.26 + 1.00i)T + (0.222 + 0.974i)T^{2}
37 1+(0.268−0.556i)T+(−0.623−0.781i)T2 1 + (0.268 - 0.556i)T + (-0.623 - 0.781i)T^{2}
41 1+T2 1 + T^{2}
43 1+(−1.26+1.00i)T+(0.222−0.974i)T2 1 + (-1.26 + 1.00i)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+(−1.57−0.360i)T+(0.900+0.433i)T2 1 + (-1.57 - 0.360i)T + (0.900 + 0.433i)T^{2}
67 1+(−0.556−0.268i)T+(0.623+0.781i)T2 1 + (-0.556 - 0.268i)T + (0.623 + 0.781i)T^{2}
71 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
73 1+(−0.483+0.385i)T+(0.222−0.974i)T2 1 + (-0.483 + 0.385i)T + (0.222 - 0.974i)T^{2}
79 1+(−0.268+0.556i)T+(−0.623−0.781i)T2 1 + (-0.268 + 0.556i)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+(−1.57+0.360i)T+(0.900−0.433i)T2 1 + (-1.57 + 0.360i)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

−8.899165881763060334203911383374, −7.88400497987356771135902019543, −7.48672024357924725841629784740, −6.85740551826042833432315854811, −6.04860350983906191130208908789, −5.00650714893477179702905186032, −4.11333105819360385564572359716, −3.62442701967029038241153605060, −2.13278273125476155769238280725, −0.800422261972713066664649704950, 1.31016054440299041107434375314, 2.21223559519177904096738983261, 3.58886033954051034692267720033, 4.82799088096735540646352264618, 5.45512431409651125413961067415, 5.82828481389231086585531353608, 6.63440995374880157456025891907, 7.46787430133799321981060530066, 8.677429515708931493482146371701, 9.317256861073392824076246013254

Graph of the ZZ-function along the critical line