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Properties

Label 2-675-27.22-c1-0-12
Degree 22
Conductor 675675
Sign −0.736−0.676i-0.736 - 0.676i
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  + (0.318 − 0.267i)2-s + (−0.159 + 1.72i)3-s + (−0.317 + 1.79i)4-s + (0.409 + 0.591i)6-s + (0.229 + 1.29i)7-s + (0.795 + 1.37i)8-s + (−2.94 − 0.551i)9-s + (4.90 + 1.78i)11-s + (−3.05 − 0.834i)12-s + (0.0138 + 0.0116i)13-s + (0.419 + 0.352i)14-s + (−2.81 − 1.02i)16-s + (−1.56 + 2.71i)17-s + (−1.08 + 0.612i)18-s + (−0.208 − 0.361i)19-s + ⋯
L(s)  = 1  + (0.225 − 0.188i)2-s + (−0.0922 + 0.995i)3-s + (−0.158 + 0.899i)4-s + (0.167 + 0.241i)6-s + (0.0866 + 0.491i)7-s + (0.281 + 0.486i)8-s + (−0.982 − 0.183i)9-s + (1.47 + 0.537i)11-s + (−0.881 − 0.241i)12-s + (0.00383 + 0.00321i)13-s + (0.112 + 0.0941i)14-s + (−0.703 − 0.256i)16-s + (−0.379 + 0.658i)17-s + (−0.255 + 0.144i)18-s + (−0.0478 − 0.0829i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((−0.736−0.676i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((−0.736−0.676i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: −0.736−0.676i-0.736 - 0.676i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(76,⋅)\chi_{675} (76, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), −0.736−0.676i)(2,\ 675,\ (\ :1/2),\ -0.736 - 0.676i)

Particular Values

L(1)L(1) ≈\approx 0.526185+1.35136i0.526185 + 1.35136i
L(12)L(\frac12) ≈\approx 0.526185+1.35136i0.526185 + 1.35136i
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+(0.159−1.72i)T 1 + (0.159 - 1.72i)T
5 1 1
good2 1+(−0.318+0.267i)T+(0.347−1.96i)T2 1 + (-0.318 + 0.267i)T + (0.347 - 1.96i)T^{2}
7 1+(−0.229−1.29i)T+(−6.57+2.39i)T2 1 + (-0.229 - 1.29i)T + (-6.57 + 2.39i)T^{2}
11 1+(−4.90−1.78i)T+(8.42+7.07i)T2 1 + (-4.90 - 1.78i)T + (8.42 + 7.07i)T^{2}
13 1+(−0.0138−0.0116i)T+(2.25+12.8i)T2 1 + (-0.0138 - 0.0116i)T + (2.25 + 12.8i)T^{2}
17 1+(1.56−2.71i)T+(−8.5−14.7i)T2 1 + (1.56 - 2.71i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.208+0.361i)T+(−9.5+16.4i)T2 1 + (0.208 + 0.361i)T + (-9.5 + 16.4i)T^{2}
23 1+(−0.179+1.01i)T+(−21.6−7.86i)T2 1 + (-0.179 + 1.01i)T + (-21.6 - 7.86i)T^{2}
29 1+(5.98−5.01i)T+(5.03−28.5i)T2 1 + (5.98 - 5.01i)T + (5.03 - 28.5i)T^{2}
31 1+(−0.647+3.67i)T+(−29.1−10.6i)T2 1 + (-0.647 + 3.67i)T + (-29.1 - 10.6i)T^{2}
37 1+(−2.21+3.83i)T+(−18.5−32.0i)T2 1 + (-2.21 + 3.83i)T + (-18.5 - 32.0i)T^{2}
41 1+(2.81+2.36i)T+(7.11+40.3i)T2 1 + (2.81 + 2.36i)T + (7.11 + 40.3i)T^{2}
43 1+(7.80+2.84i)T+(32.9+27.6i)T2 1 + (7.80 + 2.84i)T + (32.9 + 27.6i)T^{2}
47 1+(−1.23−6.99i)T+(−44.1+16.0i)T2 1 + (-1.23 - 6.99i)T + (-44.1 + 16.0i)T^{2}
53 1−1.30T+53T2 1 - 1.30T + 53T^{2}
59 1+(−3.47+1.26i)T+(45.1−37.9i)T2 1 + (-3.47 + 1.26i)T + (45.1 - 37.9i)T^{2}
61 1+(−1.20−6.80i)T+(−57.3+20.8i)T2 1 + (-1.20 - 6.80i)T + (-57.3 + 20.8i)T^{2}
67 1+(−8.44−7.08i)T+(11.6+65.9i)T2 1 + (-8.44 - 7.08i)T + (11.6 + 65.9i)T^{2}
71 1+(−3.04+5.26i)T+(−35.5−61.4i)T2 1 + (-3.04 + 5.26i)T + (-35.5 - 61.4i)T^{2}
73 1+(0.273+0.473i)T+(−36.5+63.2i)T2 1 + (0.273 + 0.473i)T + (-36.5 + 63.2i)T^{2}
79 1+(−0.374+0.314i)T+(13.7−77.7i)T2 1 + (-0.374 + 0.314i)T + (13.7 - 77.7i)T^{2}
83 1+(3.53−2.96i)T+(14.4−81.7i)T2 1 + (3.53 - 2.96i)T + (14.4 - 81.7i)T^{2}
89 1+(−1.68−2.92i)T+(−44.5+77.0i)T2 1 + (-1.68 - 2.92i)T + (-44.5 + 77.0i)T^{2}
97 1+(−9.34−3.40i)T+(74.3+62.3i)T2 1 + (-9.34 - 3.40i)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

−11.00717936246322104753323230946, −9.887036133675585424427245528359, −8.982490069268780253032083174718, −8.622524780432419390697792374749, −7.35711852094336720378873776941, −6.27957913307692559685186829930, −5.13777903888452829177494697286, −4.14460962669847745671606901376, −3.57491424097319745816096588908, −2.19588921289253512907295279318, 0.75220362470651140699437670473, 1.86222458672926423336553022083, 3.57758720805954885249524263219, 4.79734081651286234769074467305, 5.87205684951414267636878342994, 6.58948285514967127893009252797, 7.22872065819903089238414300040, 8.431020474945202536338984132347, 9.260443139134693426371122439164, 10.14751668783970845519628576095

Graph of the ZZ-function along the critical line