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Properties

Label 2-675-135.124-c1-0-47
Degree 22
Conductor 675675
Sign −0.865+0.501i-0.865 + 0.501i
Analytic cond. 5.389905.38990
Root an. cond. 2.321612.32161
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  + (1.62 − 1.93i)2-s + (1.32 − 1.11i)3-s + (−0.766 − 4.34i)4-s − 4.38i·6-s + (3.01 + 0.532i)7-s + (−5.28 − 3.05i)8-s + (0.520 − 2.95i)9-s + (−5.29 + 1.92i)11-s + (−5.85 − 4.91i)12-s + (2.71 + 3.23i)13-s + (5.94 − 4.98i)14-s + (−6.23 + 2.27i)16-s + (1.43 − 0.826i)17-s + (−4.88 − 5.81i)18-s + (0.120 − 0.208i)19-s + ⋯
L(s)  = 1  + (1.15 − 1.37i)2-s + (0.766 − 0.642i)3-s + (−0.383 − 2.17i)4-s − 1.79i·6-s + (1.14 + 0.201i)7-s + (−1.86 − 1.07i)8-s + (0.173 − 0.984i)9-s + (−1.59 + 0.581i)11-s + (−1.68 − 1.41i)12-s + (0.753 + 0.898i)13-s + (1.58 − 1.33i)14-s + (−1.55 + 0.567i)16-s + (0.347 − 0.200i)17-s + (−1.15 − 1.37i)18-s + (0.0276 − 0.0479i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((−0.865+0.501i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((−0.865+0.501i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: −0.865+0.501i-0.865 + 0.501i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(124,⋅)\chi_{675} (124, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), −0.865+0.501i)(2,\ 675,\ (\ :1/2),\ -0.865 + 0.501i)

Particular Values

L(1)L(1) ≈\approx 0.912703−3.39421i0.912703 - 3.39421i
L(12)L(\frac12) ≈\approx 0.912703−3.39421i0.912703 - 3.39421i
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)
bad3 1+(−1.32+1.11i)T 1 + (-1.32 + 1.11i)T
5 1 1
good2 1+(−1.62+1.93i)T+(−0.347−1.96i)T2 1 + (-1.62 + 1.93i)T + (-0.347 - 1.96i)T^{2}
7 1+(−3.01−0.532i)T+(6.57+2.39i)T2 1 + (-3.01 - 0.532i)T + (6.57 + 2.39i)T^{2}
11 1+(5.29−1.92i)T+(8.42−7.07i)T2 1 + (5.29 - 1.92i)T + (8.42 - 7.07i)T^{2}
13 1+(−2.71−3.23i)T+(−2.25+12.8i)T2 1 + (-2.71 - 3.23i)T + (-2.25 + 12.8i)T^{2}
17 1+(−1.43+0.826i)T+(8.5−14.7i)T2 1 + (-1.43 + 0.826i)T + (8.5 - 14.7i)T^{2}
19 1+(−0.120+0.208i)T+(−9.5−16.4i)T2 1 + (-0.120 + 0.208i)T + (-9.5 - 16.4i)T^{2}
23 1+(7.34−1.29i)T+(21.6−7.86i)T2 1 + (7.34 - 1.29i)T + (21.6 - 7.86i)T^{2}
29 1+(−5.90−4.95i)T+(5.03+28.5i)T2 1 + (-5.90 - 4.95i)T + (5.03 + 28.5i)T^{2}
31 1+(−0.858−4.86i)T+(−29.1+10.6i)T2 1 + (-0.858 - 4.86i)T + (-29.1 + 10.6i)T^{2}
37 1+(−2.15+1.24i)T+(18.5−32.0i)T2 1 + (-2.15 + 1.24i)T + (18.5 - 32.0i)T^{2}
41 1+(0.109−0.0918i)T+(7.11−40.3i)T2 1 + (0.109 - 0.0918i)T + (7.11 - 40.3i)T^{2}
43 1+(−0.256−0.705i)T+(−32.9+27.6i)T2 1 + (-0.256 - 0.705i)T + (-32.9 + 27.6i)T^{2}
47 1+(4.58+0.807i)T+(44.1+16.0i)T2 1 + (4.58 + 0.807i)T + (44.1 + 16.0i)T^{2}
53 1−12.1iT−53T2 1 - 12.1iT - 53T^{2}
59 1+(4.45+1.62i)T+(45.1+37.9i)T2 1 + (4.45 + 1.62i)T + (45.1 + 37.9i)T^{2}
61 1+(−2.41+13.6i)T+(−57.3−20.8i)T2 1 + (-2.41 + 13.6i)T + (-57.3 - 20.8i)T^{2}
67 1+(4.73+5.64i)T+(−11.6+65.9i)T2 1 + (4.73 + 5.64i)T + (-11.6 + 65.9i)T^{2}
71 1+(−2.45−4.24i)T+(−35.5+61.4i)T2 1 + (-2.45 - 4.24i)T + (-35.5 + 61.4i)T^{2}
73 1+(−0.196−0.113i)T+(36.5+63.2i)T2 1 + (-0.196 - 0.113i)T + (36.5 + 63.2i)T^{2}
79 1+(−7.53−6.32i)T+(13.7+77.7i)T2 1 + (-7.53 - 6.32i)T + (13.7 + 77.7i)T^{2}
83 1+(−5.69+6.78i)T+(−14.4−81.7i)T2 1 + (-5.69 + 6.78i)T + (-14.4 - 81.7i)T^{2}
89 1+(−3.33+5.76i)T+(−44.5−77.0i)T2 1 + (-3.33 + 5.76i)T + (-44.5 - 77.0i)T^{2}
97 1+(3.26+8.95i)T+(−74.3+62.3i)T2 1 + (3.26 + 8.95i)T + (-74.3 + 62.3i)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

−10.42725738677588412562146941051, −9.537835208141581391551842581101, −8.409173607697516748365266946613, −7.65981712342045639784283407560, −6.31470314610668691531154384607, −5.17927980558880865999322663180, −4.41627608687596871065824102618, −3.25350824375078114235048697771, −2.24135412528041975636787624262, −1.48445222708352362642519633307, 2.59296822880133799517056746840, 3.69112664571016319352659580784, 4.58880572455867920206105068126, 5.36349336157572243850622552079, 6.10045618980844807488081763959, 7.66779727235471810849593044651, 8.112207589054028994927766709683, 8.376615474006188502557979786107, 10.04678566461786229905750285130, 10.77271000708974388632975243310

Graph of the ZZ-function along the critical line