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Properties

Label 2-3724-532.11-c0-0-0
Degree 22
Conductor 37243724
Sign 0.983+0.182i0.983 + 0.182i
Analytic cond. 1.858511.85851
Root an. cond. 1.363271.36327
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  + i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 − 0.866i)6-s − i·8-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (−0.5 − 0.866i)13-s + 16-s + (−0.5 − 0.866i)17-s + i·19-s + (−0.5 + 0.866i)22-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)24-s − 25-s + (0.866 − 0.5i)26-s + ⋯
L(s)  = 1  + i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 − 0.866i)6-s − i·8-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (−0.5 − 0.866i)13-s + 16-s + (−0.5 − 0.866i)17-s + i·19-s + (−0.5 + 0.866i)22-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)24-s − 25-s + (0.866 − 0.5i)26-s + ⋯

Functional equation

Λ(s)=(3724s/2 ΓC(s) L(s)=((0.983+0.182i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3724s/2 ΓC(s) L(s)=((0.983+0.182i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37243724    =    22⋅72⋅192^{2} \cdot 7^{2} \cdot 19
Sign: 0.983+0.182i0.983 + 0.182i
Analytic conductor: 1.858511.85851
Root analytic conductor: 1.363271.36327
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3724(3203,⋅)\chi_{3724} (3203, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3724, ( :0), 0.983+0.182i)(2,\ 3724,\ (\ :0),\ 0.983 + 0.182i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.43922956390.4392295639
L(12)L(\frac12) ≈\approx 0.43922956390.4392295639
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−iT 1 - iT
7 1 1
19 1−iT 1 - iT
good3 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
5 1+T2 1 + T^{2}
11 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
13 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
17 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
29 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
31 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
37 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
41 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
43 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
47 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
53 1+T2 1 + T^{2}
59 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
61 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1−T2 1 - T^{2}
71 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
73 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
97 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)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

−8.452835346745081115083108489485, −7.925133043039206103061765521945, −7.08765356738598637752699838309, −6.36620104429970610590609122435, −5.66336607753320709288782514045, −5.11123936771592119183489251876, −4.31015622373403251912049788081, −3.65369288460554169556934239256, −2.11525661783365455683824574787, −0.31312699152461624148476796347, 1.19116476975732416251713516843, 2.02035688785779943015252161692, 3.21979818444840351160258814261, 4.08278202384591875999618649238, 4.79714441016903614037997862214, 5.78589952608206587564666818171, 6.36528851443485811562798285468, 7.10644488464723404205566703346, 8.187981378069398338861974893220, 8.842697116992846785043981322275

Graph of the ZZ-function along the critical line