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Properties

Label 2-2320-580.239-c0-0-2
Degree 22
Conductor 23202320
Sign 0.226−0.974i0.226 - 0.974i
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  + (1.75 + 0.846i)3-s + (0.222 + 0.974i)5-s + (−0.781 − 0.376i)7-s + (1.74 + 2.19i)9-s + (−0.433 + 1.90i)15-s + (−1.05 − 1.32i)21-s + (0.433 − 1.90i)23-s + (−0.900 + 0.433i)25-s + (0.781 + 3.42i)27-s + (−0.900 − 0.433i)29-s + (0.193 − 0.846i)35-s − 1.24·41-s + (0.347 − 1.52i)43-s + (−1.74 + 2.19i)45-s + (−0.541 + 0.678i)47-s + ⋯
L(s)  = 1  + (1.75 + 0.846i)3-s + (0.222 + 0.974i)5-s + (−0.781 − 0.376i)7-s + (1.74 + 2.19i)9-s + (−0.433 + 1.90i)15-s + (−1.05 − 1.32i)21-s + (0.433 − 1.90i)23-s + (−0.900 + 0.433i)25-s + (0.781 + 3.42i)27-s + (−0.900 − 0.433i)29-s + (0.193 − 0.846i)35-s − 1.24·41-s + (0.347 − 1.52i)43-s + (−1.74 + 2.19i)45-s + (−0.541 + 0.678i)47-s + ⋯

Functional equation

Λ(s)=(2320s/2 ΓC(s) L(s)=((0.226−0.974i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2320s/2 ΓC(s) L(s)=((0.226−0.974i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23202320    =    24⋅5⋅292^{4} \cdot 5 \cdot 29
Sign: 0.226−0.974i0.226 - 0.974i
Analytic conductor: 1.157831.15783
Root analytic conductor: 1.076021.07602
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2320(239,⋅)\chi_{2320} (239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2320, ( :0), 0.226−0.974i)(2,\ 2320,\ (\ :0),\ 0.226 - 0.974i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.0787554002.078755400
L(12)L(\frac12) ≈\approx 2.0787554002.078755400
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+(−0.222−0.974i)T 1 + (-0.222 - 0.974i)T
29 1+(0.900+0.433i)T 1 + (0.900 + 0.433i)T
good3 1+(−1.75−0.846i)T+(0.623+0.781i)T2 1 + (-1.75 - 0.846i)T + (0.623 + 0.781i)T^{2}
7 1+(0.781+0.376i)T+(0.623+0.781i)T2 1 + (0.781 + 0.376i)T + (0.623 + 0.781i)T^{2}
11 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
13 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
17 1−T2 1 - T^{2}
19 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
23 1+(−0.433+1.90i)T+(−0.900−0.433i)T2 1 + (-0.433 + 1.90i)T + (-0.900 - 0.433i)T^{2}
31 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
37 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
41 1+1.24T+T2 1 + 1.24T + T^{2}
43 1+(−0.347+1.52i)T+(−0.900−0.433i)T2 1 + (-0.347 + 1.52i)T + (-0.900 - 0.433i)T^{2}
47 1+(0.541−0.678i)T+(−0.222−0.974i)T2 1 + (0.541 - 0.678i)T + (-0.222 - 0.974i)T^{2}
53 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
59 1−T2 1 - T^{2}
61 1+(−1.80−0.867i)T+(0.623+0.781i)T2 1 + (-1.80 - 0.867i)T + (0.623 + 0.781i)T^{2}
67 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
71 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
73 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
79 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
83 1+(−1.40+0.678i)T+(0.623−0.781i)T2 1 + (-1.40 + 0.678i)T + (0.623 - 0.781i)T^{2}
89 1+(−0.400−1.75i)T+(−0.900+0.433i)T2 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2}
97 1+(−0.623+0.781i)T2 1 + (-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

−9.372635033935129906098830092188, −8.678128259870180868608651082504, −7.949046066587398154429994363466, −7.10929390648731498569757676176, −6.55434932761866322294752457794, −5.22063528714409003568273290602, −4.12827093668923632345646450706, −3.55720613720863023948916015080, −2.77758256141876552961259815313, −2.09442953980796103574932144286, 1.30945175201873503891558579993, 2.12582616272476739933106778804, 3.22576818148475464953404473980, 3.75547003643967298911903765274, 5.02680427488514544905490608952, 6.06271479623477986271853995061, 6.91414942428982925845908993551, 7.66364453829863136401297818461, 8.309404182603510500381358863611, 9.002641016278793909118048018871

Graph of the ZZ-function along the critical line