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Properties

Label 2-532-133.37-c0-0-1
Degree 22
Conductor 532532
Sign 0.895+0.444i0.895 + 0.444i
Analytic cond. 0.2655020.265502
Root an. cond. 0.5152690.515269
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  + (0.5 − 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s + (−1 − 1.73i)11-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)35-s − 43-s + (0.499 + 0.866i)45-s + (−1 + 1.73i)47-s + 49-s − 1.99·55-s + (−1 + 1.73i)61-s + (−0.5 + 0.866i)63-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s + (−1 − 1.73i)11-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)35-s − 43-s + (0.499 + 0.866i)45-s + (−1 + 1.73i)47-s + 49-s − 1.99·55-s + (−1 + 1.73i)61-s + (−0.5 + 0.866i)63-s + ⋯

Functional equation

Λ(s)=(532s/2 ΓC(s) L(s)=((0.895+0.444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(532s/2 ΓC(s) L(s)=((0.895+0.444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 532532    =    22⋅7⋅192^{2} \cdot 7 \cdot 19
Sign: 0.895+0.444i0.895 + 0.444i
Analytic conductor: 0.2655020.265502
Root analytic conductor: 0.5152690.515269
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ532(37,⋅)\chi_{532} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 532, ( :0), 0.895+0.444i)(2,\ 532,\ (\ :0),\ 0.895 + 0.444i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.97732398550.9773239855
L(12)L(\frac12) ≈\approx 0.97732398550.9773239855
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
7 1−T 1 - T
19 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good3 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
5 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
11 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
13 1−T2 1 - 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+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+T+T2 1 + T + T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1−T2 1 - 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

−10.80586181852080632349541324149, −10.43417693083198184373789194958, −8.918145196570350113020173579221, −8.297768829904437201319863794593, −7.84483887038179014517738524124, −6.02470916889726814118017157017, −5.44669032102847998337813284477, −4.57714236045013349233443617036, −2.97858536469963990867659022587, −1.54675859337263129998469221624, 2.02971190226874062559952940153, 3.07235717316408020694215280717, 4.66490704319140395632118686786, 5.43626596634302760288955360460, 6.77891612716172954239535247220, 7.33882750159943203159656852194, 8.431050937786117325797763253107, 9.564291806996441187246636934588, 10.16644384465903336371900381510, 11.13959054975195862916308810271

Graph of the ZZ-function along the critical line