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Properties

Label 2-1872-13.9-c1-0-18
Degree 22
Conductor 18721872
Sign 0.923−0.384i0.923 - 0.384i
Analytic cond. 14.947914.9479
Root an. cond. 3.866263.86626
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.56·5-s + (0.280 + 0.486i)7-s + (1 − 1.73i)11-s + (0.5 + 3.57i)13-s + (−0.780 − 1.35i)17-s + (3.56 + 6.16i)19-s + (−1 + 1.73i)23-s + 7.68·25-s + (3.34 − 5.78i)29-s − 2.56·31-s + (1 + 1.73i)35-s + (−3.78 + 6.54i)37-s + (−0.780 + 1.35i)41-s + (2.28 + 3.95i)43-s + 8.24·47-s + ⋯
L(s)  = 1  + 1.59·5-s + (0.106 + 0.183i)7-s + (0.301 − 0.522i)11-s + (0.138 + 0.990i)13-s + (−0.189 − 0.327i)17-s + (0.817 + 1.41i)19-s + (−0.208 + 0.361i)23-s + 1.53·25-s + (0.620 − 1.07i)29-s − 0.460·31-s + (0.169 + 0.292i)35-s + (−0.621 + 1.07i)37-s + (−0.121 + 0.211i)41-s + (0.347 + 0.602i)43-s + 1.20·47-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=((0.923−0.384i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+1/2) L(s)=((0.923−0.384i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 0.923−0.384i0.923 - 0.384i
Analytic conductor: 14.947914.9479
Root analytic conductor: 3.866263.86626
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1872(1153,⋅)\chi_{1872} (1153, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1872, ( :1/2), 0.923−0.384i)(2,\ 1872,\ (\ :1/2),\ 0.923 - 0.384i)

Particular Values

L(1)L(1) ≈\approx 2.4976352492.497635249
L(12)L(\frac12) ≈\approx 2.4976352492.497635249
L(32)L(\frac{3}{2}) 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
3 1 1
13 1+(−0.5−3.57i)T 1 + (-0.5 - 3.57i)T
good5 1−3.56T+5T2 1 - 3.56T + 5T^{2}
7 1+(−0.280−0.486i)T+(−3.5+6.06i)T2 1 + (-0.280 - 0.486i)T + (-3.5 + 6.06i)T^{2}
11 1+(−1+1.73i)T+(−5.5−9.52i)T2 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2}
17 1+(0.780+1.35i)T+(−8.5+14.7i)T2 1 + (0.780 + 1.35i)T + (-8.5 + 14.7i)T^{2}
19 1+(−3.56−6.16i)T+(−9.5+16.4i)T2 1 + (-3.56 - 6.16i)T + (-9.5 + 16.4i)T^{2}
23 1+(1−1.73i)T+(−11.5−19.9i)T2 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3.34+5.78i)T+(−14.5−25.1i)T2 1 + (-3.34 + 5.78i)T + (-14.5 - 25.1i)T^{2}
31 1+2.56T+31T2 1 + 2.56T + 31T^{2}
37 1+(3.78−6.54i)T+(−18.5−32.0i)T2 1 + (3.78 - 6.54i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.780−1.35i)T+(−20.5−35.5i)T2 1 + (0.780 - 1.35i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2.28−3.95i)T+(−21.5+37.2i)T2 1 + (-2.28 - 3.95i)T + (-21.5 + 37.2i)T^{2}
47 1−8.24T+47T2 1 - 8.24T + 47T^{2}
53 1−0.684T+53T2 1 - 0.684T + 53T^{2}
59 1+(−1.43−2.49i)T+(−29.5+51.0i)T2 1 + (-1.43 - 2.49i)T + (-29.5 + 51.0i)T^{2}
61 1+(1.93+3.35i)T+(−30.5+52.8i)T2 1 + (1.93 + 3.35i)T + (-30.5 + 52.8i)T^{2}
67 1+(−2.28+3.95i)T+(−33.5−58.0i)T2 1 + (-2.28 + 3.95i)T + (-33.5 - 58.0i)T^{2}
71 1+(7+12.1i)T+(−35.5+61.4i)T2 1 + (7 + 12.1i)T + (-35.5 + 61.4i)T^{2}
73 1+10.1T+73T2 1 + 10.1T + 73T^{2}
79 1+5.43T+79T2 1 + 5.43T + 79T^{2}
83 1+0.876T+83T2 1 + 0.876T + 83T^{2}
89 1+(−2.43+4.22i)T+(−44.5−77.0i)T2 1 + (-2.43 + 4.22i)T + (-44.5 - 77.0i)T^{2}
97 1+(−4.28−7.41i)T+(−48.5+84.0i)T2 1 + (-4.28 - 7.41i)T + (-48.5 + 84.0i)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

−9.325103184043324238972951115550, −8.720609449557943700477363563860, −7.73043287595750612361067722461, −6.68840465760489197477717599095, −6.01570562172815349605657951438, −5.48304244609786335471487191377, −4.42263564723302872281811982381, −3.27499667462837918667664030543, −2.14575147941085599467771918985, −1.34604065011123697457829307571, 1.04632549379668873469767006555, 2.17561951429215874973104856902, 3.03440136362793818258010634424, 4.35725988164092933301003533264, 5.35807984818000094637297283279, 5.80190402727421685797123992296, 6.86740871666235156099354333530, 7.38289184339981563565167438414, 8.747803213252725408541094553106, 9.087803291007023319471308564400

Graph of the ZZ-function along the critical line