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Properties

Label 2-189-189.5-c1-0-11
Degree 22
Conductor 189189
Sign 0.679−0.733i0.679 - 0.733i
Analytic cond. 1.509171.50917
Root an. cond. 1.228481.22848
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  + (1.95 − 0.345i)2-s + (−1.39 + 1.02i)3-s + (1.83 − 0.667i)4-s + (−0.649 + 3.68i)5-s + (−2.37 + 2.49i)6-s + (2.58 − 0.552i)7-s + (−0.0842 + 0.0486i)8-s + (0.891 − 2.86i)9-s + 7.43i·10-s + (0.526 − 0.0928i)11-s + (−1.87 + 2.81i)12-s + (3.22 − 3.84i)13-s + (4.87 − 1.97i)14-s + (−2.87 − 5.80i)15-s + (−3.13 + 2.63i)16-s − 4.53·17-s + ⋯
L(s)  = 1  + (1.38 − 0.244i)2-s + (−0.805 + 0.592i)3-s + (0.916 − 0.333i)4-s + (−0.290 + 1.64i)5-s + (−0.970 + 1.01i)6-s + (0.977 − 0.208i)7-s + (−0.0297 + 0.0171i)8-s + (0.297 − 0.954i)9-s + 2.35i·10-s + (0.158 − 0.0280i)11-s + (−0.540 + 0.812i)12-s + (0.894 − 1.06i)13-s + (1.30 − 0.527i)14-s + (−0.742 − 1.49i)15-s + (−0.784 + 0.658i)16-s − 1.10·17-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=((0.679−0.733i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+1/2) L(s)=((0.679−0.733i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: 0.679−0.733i0.679 - 0.733i
Analytic conductor: 1.509171.50917
Root analytic conductor: 1.228481.22848
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ189(5,⋅)\chi_{189} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 189, ( :1/2), 0.679−0.733i)(2,\ 189,\ (\ :1/2),\ 0.679 - 0.733i)

Particular Values

L(1)L(1) ≈\approx 1.67845+0.733119i1.67845 + 0.733119i
L(12)L(\frac12) ≈\approx 1.67845+0.733119i1.67845 + 0.733119i
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+(1.39−1.02i)T 1 + (1.39 - 1.02i)T
7 1+(−2.58+0.552i)T 1 + (-2.58 + 0.552i)T
good2 1+(−1.95+0.345i)T+(1.87−0.684i)T2 1 + (-1.95 + 0.345i)T + (1.87 - 0.684i)T^{2}
5 1+(0.649−3.68i)T+(−4.69−1.71i)T2 1 + (0.649 - 3.68i)T + (-4.69 - 1.71i)T^{2}
11 1+(−0.526+0.0928i)T+(10.3−3.76i)T2 1 + (-0.526 + 0.0928i)T + (10.3 - 3.76i)T^{2}
13 1+(−3.22+3.84i)T+(−2.25−12.8i)T2 1 + (-3.22 + 3.84i)T + (-2.25 - 12.8i)T^{2}
17 1+4.53T+17T2 1 + 4.53T + 17T^{2}
19 1+1.20iT−19T2 1 + 1.20iT - 19T^{2}
23 1+(−5.35+6.37i)T+(−3.99−22.6i)T2 1 + (-5.35 + 6.37i)T + (-3.99 - 22.6i)T^{2}
29 1+(−2.30−2.75i)T+(−5.03+28.5i)T2 1 + (-2.30 - 2.75i)T + (-5.03 + 28.5i)T^{2}
31 1+(0.0334+0.0920i)T+(−23.7+19.9i)T2 1 + (0.0334 + 0.0920i)T + (-23.7 + 19.9i)T^{2}
37 1+(0.267+0.463i)T+(−18.5+32.0i)T2 1 + (0.267 + 0.463i)T + (-18.5 + 32.0i)T^{2}
41 1+(5.79+4.85i)T+(7.11+40.3i)T2 1 + (5.79 + 4.85i)T + (7.11 + 40.3i)T^{2}
43 1+(0.0316+0.0115i)T+(32.9+27.6i)T2 1 + (0.0316 + 0.0115i)T + (32.9 + 27.6i)T^{2}
47 1+(−5.19−1.88i)T+(36.0+30.2i)T2 1 + (-5.19 - 1.88i)T + (36.0 + 30.2i)T^{2}
53 1+(8.57−4.95i)T+(26.5−45.8i)T2 1 + (8.57 - 4.95i)T + (26.5 - 45.8i)T^{2}
59 1+(−0.583−0.489i)T+(10.2+58.1i)T2 1 + (-0.583 - 0.489i)T + (10.2 + 58.1i)T^{2}
61 1+(−2.21+6.08i)T+(−46.7−39.2i)T2 1 + (-2.21 + 6.08i)T + (-46.7 - 39.2i)T^{2}
67 1+(1.14−6.48i)T+(−62.9−22.9i)T2 1 + (1.14 - 6.48i)T + (-62.9 - 22.9i)T^{2}
71 1+(−3.13−1.80i)T+(35.5+61.4i)T2 1 + (-3.13 - 1.80i)T + (35.5 + 61.4i)T^{2}
73 1+(5.76+3.32i)T+(36.5+63.2i)T2 1 + (5.76 + 3.32i)T + (36.5 + 63.2i)T^{2}
79 1+(0.909+5.15i)T+(−74.2+27.0i)T2 1 + (0.909 + 5.15i)T + (-74.2 + 27.0i)T^{2}
83 1+(1.39−1.16i)T+(14.4−81.7i)T2 1 + (1.39 - 1.16i)T + (14.4 - 81.7i)T^{2}
89 1+9.21T+89T2 1 + 9.21T + 89T^{2}
97 1+(4.37−12.0i)T+(−74.3−62.3i)T2 1 + (4.37 - 12.0i)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

−12.64002798308935370244369479492, −11.55347890219902373874552343884, −10.84718758816533908074842536149, −10.62996727545444563232872481578, −8.668080393374953276293076636931, −6.99196275252002892636718298302, −6.19536102595189021127701849736, −5.00537260043365436046636038293, −3.97269548795570768099179771681, −2.87481072522384202088931738819, 1.54346126567714391196270670820, 4.22314541514449113764600441404, 4.88216914289051433189769410904, 5.72931026493107742735202246866, 6.89666078928681928103056481542, 8.262792134062950332870449504144, 9.194810595198723040598192455965, 11.32954548052963590393516753629, 11.65917607017913961268209899644, 12.57346584960623238992983206248

Graph of the ZZ-function along the critical line