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Properties

Label 2-15e2-225.106-c1-0-15
Degree 22
Conductor 225225
Sign 0.658−0.752i0.658 - 0.752i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
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  + (2.10 + 0.447i)2-s + (1.01 + 1.40i)3-s + (2.39 + 1.06i)4-s + (−1.89 − 1.19i)5-s + (1.50 + 3.40i)6-s + (−0.107 + 0.185i)7-s + (1.08 + 0.786i)8-s + (−0.937 + 2.84i)9-s + (−3.44 − 3.35i)10-s + (0.814 + 0.173i)11-s + (0.936 + 4.44i)12-s + (5.35 − 1.13i)13-s + (−0.308 + 0.343i)14-s + (−0.244 − 3.86i)15-s + (−1.58 − 1.75i)16-s + (−4.59 − 3.33i)17-s + ⋯
L(s)  = 1  + (1.48 + 0.316i)2-s + (0.586 + 0.810i)3-s + (1.19 + 0.533i)4-s + (−0.845 − 0.534i)5-s + (0.615 + 1.38i)6-s + (−0.0405 + 0.0702i)7-s + (0.382 + 0.278i)8-s + (−0.312 + 0.949i)9-s + (−1.08 − 1.06i)10-s + (0.245 + 0.0521i)11-s + (0.270 + 1.28i)12-s + (1.48 − 0.315i)13-s + (−0.0825 + 0.0916i)14-s + (−0.0630 − 0.998i)15-s + (−0.395 − 0.439i)16-s + (−1.11 − 0.809i)17-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((0.658−0.752i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((0.658−0.752i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: 0.658−0.752i0.658 - 0.752i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ225(106,⋅)\chi_{225} (106, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 225, ( :1/2), 0.658−0.752i)(2,\ 225,\ (\ :1/2),\ 0.658 - 0.752i)

Particular Values

L(1)L(1) ≈\approx 2.37507+1.07798i2.37507 + 1.07798i
L(12)L(\frac12) ≈\approx 2.37507+1.07798i2.37507 + 1.07798i
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+(−1.01−1.40i)T 1 + (-1.01 - 1.40i)T
5 1+(1.89+1.19i)T 1 + (1.89 + 1.19i)T
good2 1+(−2.10−0.447i)T+(1.82+0.813i)T2 1 + (-2.10 - 0.447i)T + (1.82 + 0.813i)T^{2}
7 1+(0.107−0.185i)T+(−3.5−6.06i)T2 1 + (0.107 - 0.185i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.814−0.173i)T+(10.0+4.47i)T2 1 + (-0.814 - 0.173i)T + (10.0 + 4.47i)T^{2}
13 1+(−5.35+1.13i)T+(11.8−5.28i)T2 1 + (-5.35 + 1.13i)T + (11.8 - 5.28i)T^{2}
17 1+(4.59+3.33i)T+(5.25+16.1i)T2 1 + (4.59 + 3.33i)T + (5.25 + 16.1i)T^{2}
19 1+(3.93+2.85i)T+(5.87+18.0i)T2 1 + (3.93 + 2.85i)T + (5.87 + 18.0i)T^{2}
23 1+(0.383−0.425i)T+(−2.40−22.8i)T2 1 + (0.383 - 0.425i)T + (-2.40 - 22.8i)T^{2}
29 1+(−0.0969−0.922i)T+(−28.3+6.02i)T2 1 + (-0.0969 - 0.922i)T + (-28.3 + 6.02i)T^{2}
31 1+(0.107−1.02i)T+(−30.3−6.44i)T2 1 + (0.107 - 1.02i)T + (-30.3 - 6.44i)T^{2}
37 1+(2.04−6.28i)T+(−29.9−21.7i)T2 1 + (2.04 - 6.28i)T + (-29.9 - 21.7i)T^{2}
41 1+(−9.61+2.04i)T+(37.4−16.6i)T2 1 + (-9.61 + 2.04i)T + (37.4 - 16.6i)T^{2}
43 1+(−2.16+3.75i)T+(−21.5−37.2i)T2 1 + (-2.16 + 3.75i)T + (-21.5 - 37.2i)T^{2}
47 1+(−0.270−2.57i)T+(−45.9+9.77i)T2 1 + (-0.270 - 2.57i)T + (-45.9 + 9.77i)T^{2}
53 1+(7.56−5.49i)T+(16.3−50.4i)T2 1 + (7.56 - 5.49i)T + (16.3 - 50.4i)T^{2}
59 1+(9.56−2.03i)T+(53.8−23.9i)T2 1 + (9.56 - 2.03i)T + (53.8 - 23.9i)T^{2}
61 1+(−14.7−3.13i)T+(55.7+24.8i)T2 1 + (-14.7 - 3.13i)T + (55.7 + 24.8i)T^{2}
67 1+(1.12−10.7i)T+(−65.5−13.9i)T2 1 + (1.12 - 10.7i)T + (-65.5 - 13.9i)T^{2}
71 1+(2.48−1.80i)T+(21.9−67.5i)T2 1 + (2.48 - 1.80i)T + (21.9 - 67.5i)T^{2}
73 1+(−1.33−4.10i)T+(−59.0+42.9i)T2 1 + (-1.33 - 4.10i)T + (-59.0 + 42.9i)T^{2}
79 1+(0.869+8.27i)T+(−77.2+16.4i)T2 1 + (0.869 + 8.27i)T + (-77.2 + 16.4i)T^{2}
83 1+(3.17−1.41i)T+(55.5−61.6i)T2 1 + (3.17 - 1.41i)T + (55.5 - 61.6i)T^{2}
89 1+(2.45+7.54i)T+(−72.0+52.3i)T2 1 + (2.45 + 7.54i)T + (-72.0 + 52.3i)T^{2}
97 1+(1.06+10.1i)T+(−94.8+20.1i)T2 1 + (1.06 + 10.1i)T + (-94.8 + 20.1i)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

−12.71402306933542163897872989560, −11.49279258656865708259954226988, −10.86896695446160898456299052319, −9.175081435038938691423567644847, −8.515161188676313590200290410199, −7.16980610755022199789783312546, −5.87193113080980359132027466344, −4.62544007598209858971730743386, −4.07196259693106199393775863402, −2.92084570496842401514652746858, 2.13328443864075057126307409750, 3.58191266634263940093198417064, 4.14237762546584046497621935566, 6.12841978404699464040132397166, 6.62690861996511198893296476114, 8.044493081460646699804994742694, 8.889063979448834963214881902139, 10.84497675154908275935576738537, 11.37353846323086792481975100627, 12.44443457005168842791118750300

Graph of the ZZ-function along the critical line