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Properties

Label 2-3328-104.85-c0-0-1
Degree 22
Conductor 33283328
Sign 0.283+0.958i0.283 + 0.958i
Analytic cond. 1.660881.66088
Root an. cond. 1.288751.28875
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  + (0.366 − 0.366i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)13-s + (0.866 − 0.5i)17-s + 0.732i·25-s + (−1.5 − 0.866i)29-s + (−1.86 + 0.5i)37-s + (1.86 − 0.5i)41-s + (−0.5 − 0.133i)45-s + (−0.866 − 0.5i)49-s − i·53-s + (0.866 − 0.5i)61-s + (−0.133 − 0.5i)65-s + (1.36 − 1.36i)73-s + (−0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + (0.366 − 0.366i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)13-s + (0.866 − 0.5i)17-s + 0.732i·25-s + (−1.5 − 0.866i)29-s + (−1.86 + 0.5i)37-s + (1.86 − 0.5i)41-s + (−0.5 − 0.133i)45-s + (−0.866 − 0.5i)49-s − i·53-s + (0.866 − 0.5i)61-s + (−0.133 − 0.5i)65-s + (1.36 − 1.36i)73-s + (−0.499 + 0.866i)81-s + ⋯

Functional equation

Λ(s)=(3328s/2 ΓC(s) L(s)=((0.283+0.958i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3328s/2 ΓC(s) L(s)=((0.283+0.958i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33283328    =    28⋅132^{8} \cdot 13
Sign: 0.283+0.958i0.283 + 0.958i
Analytic conductor: 1.660881.66088
Root analytic conductor: 1.288751.28875
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3328(2945,⋅)\chi_{3328} (2945, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3328, ( :0), 0.283+0.958i)(2,\ 3328,\ (\ :0),\ 0.283 + 0.958i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2407979711.240797971
L(12)L(\frac12) ≈\approx 1.2407979711.240797971
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
13 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+(−0.366+0.366i)T−iT2 1 + (-0.366 + 0.366i)T - iT^{2}
7 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
11 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
17 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
19 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
23 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
31 1+iT2 1 + iT^{2}
37 1+(1.86−0.5i)T+(0.866−0.5i)T2 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2}
41 1+(−1.86+0.5i)T+(0.866−0.5i)T2 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2}
43 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
47 1−iT2 1 - iT^{2}
53 1+iT−T2 1 + iT - T^{2}
59 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
61 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
67 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
71 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
73 1+(−1.36+1.36i)T−iT2 1 + (-1.36 + 1.36i)T - iT^{2}
79 1+T2 1 + T^{2}
83 1−iT2 1 - iT^{2}
89 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
97 1+(−0.366+1.36i)T+(−0.866−0.5i)T2 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)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

−8.711398981065079343866128561323, −7.960162431337252185807076849113, −7.22285804697962889266174896650, −6.26485077718987502282258671847, −5.60157827307155737785397712986, −5.07650242769767254976628759390, −3.72048342350847623366817525846, −3.27027318632784256529949984770, −2.00359710738641751222314370834, −0.76009050882679437910665504439, 1.55275188061677362922692401739, 2.41165000993393497668636566278, 3.43123527672051933819983800130, 4.28604815814138452805054339504, 5.33594934651065302878081337204, 5.87089796064282651685902778272, 6.71650543991791644208186699983, 7.50150163613653361597015490515, 8.204138262500032275404458138142, 8.967290125692116922375920852404

Graph of the ZZ-function along the critical line