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Properties

Label 2-2523-87.71-c0-0-3
Degree 22
Conductor 25232523
Sign 0.447+0.894i0.447 + 0.894i
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  + (−0.781 − 0.623i)3-s + (−0.623 − 0.781i)4-s + (−1.00 + 1.26i)7-s + (0.222 + 0.974i)9-s + 0.999i·12-s + (0.137 − 0.602i)13-s + (−0.222 + 0.974i)16-s + (0.483 − 0.385i)19-s + (1.57 − 0.360i)21-s + (0.623 + 0.781i)25-s + (0.433 − 0.900i)27-s + 1.61·28-s + (0.702 − 1.45i)31-s + (0.623 − 0.781i)36-s + (0.602 − 0.137i)37-s + ⋯
L(s)  = 1  + (−0.781 − 0.623i)3-s + (−0.623 − 0.781i)4-s + (−1.00 + 1.26i)7-s + (0.222 + 0.974i)9-s + 0.999i·12-s + (0.137 − 0.602i)13-s + (−0.222 + 0.974i)16-s + (0.483 − 0.385i)19-s + (1.57 − 0.360i)21-s + (0.623 + 0.781i)25-s + (0.433 − 0.900i)27-s + 1.61·28-s + (0.702 − 1.45i)31-s + (0.623 − 0.781i)36-s + (0.602 − 0.137i)37-s + ⋯

Functional equation

Λ(s)=(2523s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2523s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25232523    =    3⋅2923 \cdot 29^{2}
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 1.259141.25914
Root analytic conductor: 1.122111.12211
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2523(2333,⋅)\chi_{2523} (2333, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2523, ( :0), 0.447+0.894i)(2,\ 2523,\ (\ :0),\ 0.447 + 0.894i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.62923421130.6292342113
L(12)L(\frac12) ≈\approx 0.62923421130.6292342113
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+(0.781+0.623i)T 1 + (0.781 + 0.623i)T
29 1 1
good2 1+(0.623+0.781i)T2 1 + (0.623 + 0.781i)T^{2}
5 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
7 1+(1.00−1.26i)T+(−0.222−0.974i)T2 1 + (1.00 - 1.26i)T + (-0.222 - 0.974i)T^{2}
11 1+(−0.900−0.433i)T2 1 + (-0.900 - 0.433i)T^{2}
13 1+(−0.137+0.602i)T+(−0.900−0.433i)T2 1 + (-0.137 + 0.602i)T + (-0.900 - 0.433i)T^{2}
17 1+T2 1 + T^{2}
19 1+(−0.483+0.385i)T+(0.222−0.974i)T2 1 + (-0.483 + 0.385i)T + (0.222 - 0.974i)T^{2}
23 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
31 1+(−0.702+1.45i)T+(−0.623−0.781i)T2 1 + (-0.702 + 1.45i)T + (-0.623 - 0.781i)T^{2}
37 1+(−0.602+0.137i)T+(0.900−0.433i)T2 1 + (-0.602 + 0.137i)T + (0.900 - 0.433i)T^{2}
41 1+T2 1 + T^{2}
43 1+(0.702+1.45i)T+(−0.623+0.781i)T2 1 + (0.702 + 1.45i)T + (-0.623 + 0.781i)T^{2}
47 1+(−0.900−0.433i)T2 1 + (-0.900 - 0.433i)T^{2}
53 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
59 1−T2 1 - T^{2}
61 1+(−1.26−1.00i)T+(0.222+0.974i)T2 1 + (-1.26 - 1.00i)T + (0.222 + 0.974i)T^{2}
67 1+(−0.137−0.602i)T+(−0.900+0.433i)T2 1 + (-0.137 - 0.602i)T + (-0.900 + 0.433i)T^{2}
71 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
73 1+(0.268+0.556i)T+(−0.623+0.781i)T2 1 + (0.268 + 0.556i)T + (-0.623 + 0.781i)T^{2}
79 1+(0.602−0.137i)T+(0.900−0.433i)T2 1 + (0.602 - 0.137i)T + (0.900 - 0.433i)T^{2}
83 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
89 1+(0.623+0.781i)T2 1 + (0.623 + 0.781i)T^{2}
97 1+(−1.26+1.00i)T+(0.222−0.974i)T2 1 + (-1.26 + 1.00i)T + (0.222 - 0.974i)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

−8.983967351293250231728298147209, −8.339284996772870821817336085956, −7.24639423546378888429633266219, −6.45978986777240623344816600286, −5.66939287166789494211504389952, −5.47685306498139793594330806570, −4.41013119292383430822666637704, −3.08007443178083725859172251232, −2.05343757789164354178090807491, −0.66185113984262191023502915910, 0.907283815424059814689246507850, 3.06012684181152651747183641252, 3.68630147658889003507356410270, 4.43051368746593416092071226949, 5.04740375109040675759741820280, 6.36316521440990021713132945629, 6.75884300083555550465672662094, 7.64669904382527152178873595279, 8.567951317729646489561141178968, 9.400228770298539088746450802191

Graph of the ZZ-function along the critical line