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Properties

Label 2-2320-580.3-c0-0-0
Degree 22
Conductor 23202320
Sign 0.784−0.619i0.784 - 0.619i
Analytic cond. 1.157831.15783
Root an. cond. 1.076021.07602
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)5-s + (0.623 + 0.781i)9-s + (0.211 + 1.87i)13-s − 0.445i·17-s + (0.222 + 0.974i)25-s + (−0.781 − 0.623i)29-s + (0.541 + 0.678i)37-s + (1.33 + 1.33i)41-s − 0.999i·45-s + (0.781 − 0.623i)49-s + (−0.900 + 1.43i)53-s + (1.59 − 0.559i)61-s + (1.00 − 1.59i)65-s + (1.90 + 0.433i)73-s + (−0.222 + 0.974i)81-s + ⋯
L(s)  = 1  + (−0.781 − 0.623i)5-s + (0.623 + 0.781i)9-s + (0.211 + 1.87i)13-s − 0.445i·17-s + (0.222 + 0.974i)25-s + (−0.781 − 0.623i)29-s + (0.541 + 0.678i)37-s + (1.33 + 1.33i)41-s − 0.999i·45-s + (0.781 − 0.623i)49-s + (−0.900 + 1.43i)53-s + (1.59 − 0.559i)61-s + (1.00 − 1.59i)65-s + (1.90 + 0.433i)73-s + (−0.222 + 0.974i)81-s + ⋯

Functional equation

Λ(s)=(2320s/2 ΓC(s) L(s)=((0.784−0.619i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2320s/2 ΓC(s) L(s)=((0.784−0.619i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23202320    =    24⋅5⋅292^{4} \cdot 5 \cdot 29
Sign: 0.784−0.619i0.784 - 0.619i
Analytic conductor: 1.157831.15783
Root analytic conductor: 1.076021.07602
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2320(1743,⋅)\chi_{2320} (1743, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2320, ( :0), 0.784−0.619i)(2,\ 2320,\ (\ :0),\ 0.784 - 0.619i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0368755801.036875580
L(12)L(\frac12) ≈\approx 1.0368755801.036875580
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 1
5 1+(0.781+0.623i)T 1 + (0.781 + 0.623i)T
29 1+(0.781+0.623i)T 1 + (0.781 + 0.623i)T
good3 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
7 1+(−0.781+0.623i)T2 1 + (-0.781 + 0.623i)T^{2}
11 1+(0.974−0.222i)T2 1 + (0.974 - 0.222i)T^{2}
13 1+(−0.211−1.87i)T+(−0.974+0.222i)T2 1 + (-0.211 - 1.87i)T + (-0.974 + 0.222i)T^{2}
17 1+0.445iT−T2 1 + 0.445iT - T^{2}
19 1+(−0.781−0.623i)T2 1 + (-0.781 - 0.623i)T^{2}
23 1+(−0.433+0.900i)T2 1 + (-0.433 + 0.900i)T^{2}
31 1+(−0.433−0.900i)T2 1 + (-0.433 - 0.900i)T^{2}
37 1+(−0.541−0.678i)T+(−0.222+0.974i)T2 1 + (-0.541 - 0.678i)T + (-0.222 + 0.974i)T^{2}
41 1+(−1.33−1.33i)T+iT2 1 + (-1.33 - 1.33i)T + iT^{2}
43 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
47 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
53 1+(0.900−1.43i)T+(−0.433−0.900i)T2 1 + (0.900 - 1.43i)T + (-0.433 - 0.900i)T^{2}
59 1+T2 1 + T^{2}
61 1+(−1.59+0.559i)T+(0.781−0.623i)T2 1 + (-1.59 + 0.559i)T + (0.781 - 0.623i)T^{2}
67 1+(0.974+0.222i)T2 1 + (0.974 + 0.222i)T^{2}
71 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
73 1+(−1.90−0.433i)T+(0.900+0.433i)T2 1 + (-1.90 - 0.433i)T + (0.900 + 0.433i)T^{2}
79 1+(0.974+0.222i)T2 1 + (0.974 + 0.222i)T^{2}
83 1+(−0.781−0.623i)T2 1 + (-0.781 - 0.623i)T^{2}
89 1+(−1.05+1.68i)T+(−0.433−0.900i)T2 1 + (-1.05 + 1.68i)T + (-0.433 - 0.900i)T^{2}
97 1+(1.62−0.781i)T+(0.623−0.781i)T2 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)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

−9.330950453339572598535624550307, −8.446365777188569968629500470860, −7.72793778981727250505910773604, −7.09884278184248528900780446618, −6.24231501506850554212907137588, −5.08113324575349435818295695005, −4.41271792380512078789620648649, −3.85439692142547285669532824514, −2.41273856180707817060281610102, −1.33564074870340698000494528881, 0.808527199387539425724671641099, 2.47268035570756624707280087695, 3.54198594443047290609996824488, 3.93672083381781090002111122143, 5.21988430550303855016514660077, 6.02678347620048283737958884164, 6.88608234712598415581696774790, 7.60799848381889434301728536154, 8.157120858125783819939897285282, 9.094755600018557613281336230325

Graph of the ZZ-function along the critical line