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Properties

Label 2-3380-260.199-c0-0-3
Degree 22
Conductor 33803380
Sign −0.499+0.866i-0.499 + 0.866i
Analytic cond. 1.686831.68683
Root an. cond. 1.298781.29878
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.900 + 1.56i)3-s + (0.499 + 0.866i)4-s + i·5-s + (−1.56 + 0.900i)6-s + (−0.385 + 0.222i)7-s + 0.999i·8-s + (−1.12 − 1.94i)9-s + (−0.5 + 0.866i)10-s − 1.80·12-s − 0.445·14-s + (−1.56 − 0.900i)15-s + (−0.5 + 0.866i)16-s − 2.24i·18-s + (−0.866 + 0.499i)20-s − 0.801i·21-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.900 + 1.56i)3-s + (0.499 + 0.866i)4-s + i·5-s + (−1.56 + 0.900i)6-s + (−0.385 + 0.222i)7-s + 0.999i·8-s + (−1.12 − 1.94i)9-s + (−0.5 + 0.866i)10-s − 1.80·12-s − 0.445·14-s + (−1.56 − 0.900i)15-s + (−0.5 + 0.866i)16-s − 2.24i·18-s + (−0.866 + 0.499i)20-s − 0.801i·21-s + ⋯

Functional equation

Λ(s)=(3380s/2 ΓC(s) L(s)=((−0.499+0.866i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3380s/2 ΓC(s) L(s)=((−0.499+0.866i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33803380    =    22⋅5⋅1322^{2} \cdot 5 \cdot 13^{2}
Sign: −0.499+0.866i-0.499 + 0.866i
Analytic conductor: 1.686831.68683
Root analytic conductor: 1.298781.29878
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3380(1499,⋅)\chi_{3380} (1499, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3380, ( :0), −0.499+0.866i)(2,\ 3380,\ (\ :0),\ -0.499 + 0.866i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1644957961.164495796
L(12)L(\frac12) ≈\approx 1.1644957961.164495796
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
5 1−iT 1 - iT
13 1 1
good3 1+(0.900−1.56i)T+(−0.5−0.866i)T2 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2}
7 1+(0.385−0.222i)T+(0.5−0.866i)T2 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2}
11 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
23 1+(0.623−1.07i)T+(−0.5−0.866i)T2 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2}
29 1+(−0.900+1.56i)T+(−0.5−0.866i)T2 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2}
31 1+T2 1 + T^{2}
37 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
41 1+(−0.385−0.222i)T+(0.5+0.866i)T2 1 + (-0.385 - 0.222i)T + (0.5 + 0.866i)T^{2}
43 1+(0.623+1.07i)T+(−0.5+0.866i)T2 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2}
47 1−1.24iT−T2 1 - 1.24iT - T^{2}
53 1−T2 1 - T^{2}
59 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
61 1+(−0.222−0.385i)T+(−0.5+0.866i)T2 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2}
67 1+(−1.07−0.623i)T+(0.5+0.866i)T2 1 + (-1.07 - 0.623i)T + (0.5 + 0.866i)T^{2}
71 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
73 1+T2 1 + T^{2}
79 1−T2 1 - T^{2}
83 1−1.80iT−T2 1 - 1.80iT - T^{2}
89 1+(−1.07−0.623i)T+(0.5+0.866i)T2 1 + (-1.07 - 0.623i)T + (0.5 + 0.866i)T^{2}
97 1+(−0.5+0.866i)T2 1 + (-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

−9.622138568555005166744702573901, −8.508810752169691237740382640951, −7.61405118205600771669953088113, −6.67604840758029486624224351810, −6.04094550293886930895688259155, −5.62301049518186736211696972087, −4.68208550517152644313435500020, −3.96886677605457110447487823064, −3.36496412211232152366631603809, −2.48308321934877943424920698010, 0.57016178644690592563445887186, 1.51415988002894388174364802078, 2.33961915151079165590267788065, 3.54709859381052730583812569166, 4.76764973664758085161829698644, 5.17007727693803576070613451600, 6.13729235154775748345393095208, 6.52568533544690427915511095798, 7.29142011989877076191377474271, 8.144256571343769364385501600882

Graph of the ZZ-function along the critical line