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Properties

Label 2-245-245.103-c1-0-12
Degree 22
Conductor 245245
Sign 0.297−0.954i0.297 - 0.954i
Analytic cond. 1.956331.95633
Root an. cond. 1.398691.39869
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  + (−0.421 + 0.571i)2-s + (−0.246 + 0.0466i)3-s + (0.440 + 1.42i)4-s + (2.22 + 0.262i)5-s + (0.0773 − 0.160i)6-s + (2.64 − 0.120i)7-s + (−2.34 − 0.820i)8-s + (−2.73 + 1.07i)9-s + (−1.08 + 1.15i)10-s + (0.529 − 1.34i)11-s + (−0.175 − 0.331i)12-s + (3.28 + 0.370i)13-s + (−1.04 + 1.56i)14-s + (−0.559 + 0.0388i)15-s + (−1.01 + 0.690i)16-s + (−0.630 − 0.0235i)17-s + ⋯
L(s)  = 1  + (−0.298 + 0.404i)2-s + (−0.142 + 0.0269i)3-s + (0.220 + 0.714i)4-s + (0.993 + 0.117i)5-s + (0.0315 − 0.0655i)6-s + (0.998 − 0.0456i)7-s + (−0.828 − 0.289i)8-s + (−0.911 + 0.357i)9-s + (−0.343 + 0.366i)10-s + (0.159 − 0.406i)11-s + (−0.0505 − 0.0957i)12-s + (0.910 + 0.102i)13-s + (−0.279 + 0.417i)14-s + (−0.144 + 0.0100i)15-s + (−0.253 + 0.172i)16-s + (−0.152 − 0.00572i)17-s + ⋯

Functional equation

Λ(s)=(245s/2 ΓC(s) L(s)=((0.297−0.954i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(245s/2 ΓC(s+1/2) L(s)=((0.297−0.954i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 245245    =    5⋅725 \cdot 7^{2}
Sign: 0.297−0.954i0.297 - 0.954i
Analytic conductor: 1.956331.95633
Root analytic conductor: 1.398691.39869
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ245(103,⋅)\chi_{245} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 245, ( :1/2), 0.297−0.954i)(2,\ 245,\ (\ :1/2),\ 0.297 - 0.954i)

Particular Values

L(1)L(1) ≈\approx 1.02307+0.752733i1.02307 + 0.752733i
L(12)L(\frac12) ≈\approx 1.02307+0.752733i1.02307 + 0.752733i
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)
bad5 1+(−2.22−0.262i)T 1 + (-2.22 - 0.262i)T
7 1+(−2.64+0.120i)T 1 + (-2.64 + 0.120i)T
good2 1+(0.421−0.571i)T+(−0.589−1.91i)T2 1 + (0.421 - 0.571i)T + (-0.589 - 1.91i)T^{2}
3 1+(0.246−0.0466i)T+(2.79−1.09i)T2 1 + (0.246 - 0.0466i)T + (2.79 - 1.09i)T^{2}
11 1+(−0.529+1.34i)T+(−8.06−7.48i)T2 1 + (-0.529 + 1.34i)T + (-8.06 - 7.48i)T^{2}
13 1+(−3.28−0.370i)T+(12.6+2.89i)T2 1 + (-3.28 - 0.370i)T + (12.6 + 2.89i)T^{2}
17 1+(0.630+0.0235i)T+(16.9+1.27i)T2 1 + (0.630 + 0.0235i)T + (16.9 + 1.27i)T^{2}
19 1+(1.79−3.10i)T+(−9.5−16.4i)T2 1 + (1.79 - 3.10i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.0779−2.08i)T+(−22.9+1.71i)T2 1 + (-0.0779 - 2.08i)T + (-22.9 + 1.71i)T^{2}
29 1+(0.472−0.107i)T+(26.1−12.5i)T2 1 + (0.472 - 0.107i)T + (26.1 - 12.5i)T^{2}
31 1+(5.64−3.25i)T+(15.5−26.8i)T2 1 + (5.64 - 3.25i)T + (15.5 - 26.8i)T^{2}
37 1+(−6.99+3.69i)T+(20.8−30.5i)T2 1 + (-6.99 + 3.69i)T + (20.8 - 30.5i)T^{2}
41 1+(4.67+9.70i)T+(−25.5+32.0i)T2 1 + (4.67 + 9.70i)T + (-25.5 + 32.0i)T^{2}
43 1+(4.24+12.1i)T+(−33.6+26.8i)T2 1 + (4.24 + 12.1i)T + (-33.6 + 26.8i)T^{2}
47 1+(−1.53−1.13i)T+(13.8+44.9i)T2 1 + (-1.53 - 1.13i)T + (13.8 + 44.9i)T^{2}
53 1+(−1.05−0.557i)T+(29.8+43.7i)T2 1 + (-1.05 - 0.557i)T + (29.8 + 43.7i)T^{2}
59 1+(0.818+10.9i)T+(−58.3+8.79i)T2 1 + (0.818 + 10.9i)T + (-58.3 + 8.79i)T^{2}
61 1+(3.36−10.9i)T+(−50.4−34.3i)T2 1 + (3.36 - 10.9i)T + (-50.4 - 34.3i)T^{2}
67 1+(−4.90−1.31i)T+(58.0+33.5i)T2 1 + (-4.90 - 1.31i)T + (58.0 + 33.5i)T^{2}
71 1+(−1.90+8.35i)T+(−63.9−30.8i)T2 1 + (-1.90 + 8.35i)T + (-63.9 - 30.8i)T^{2}
73 1+(0.574−0.424i)T+(21.5−69.7i)T2 1 + (0.574 - 0.424i)T + (21.5 - 69.7i)T^{2}
79 1+(8.89+5.13i)T+(39.5+68.4i)T2 1 + (8.89 + 5.13i)T + (39.5 + 68.4i)T^{2}
83 1+(1.25+11.1i)T+(−80.9+18.4i)T2 1 + (1.25 + 11.1i)T + (-80.9 + 18.4i)T^{2}
89 1+(−3.16−8.07i)T+(−65.2+60.5i)T2 1 + (-3.16 - 8.07i)T + (-65.2 + 60.5i)T^{2}
97 1+(−6.86−6.86i)T+97iT2 1 + (-6.86 - 6.86i)T + 97iT^{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

−12.17327565174953096292211043285, −11.24049950565783350153514839445, −10.55493946890192005220020367474, −8.965779335543883632647788019908, −8.527546133928515235295103672279, −7.40628348131077522826873984565, −6.18638171874793030052514195854, −5.38492810982080751450320079634, −3.62975001052423597817956947000, −2.07038022591944037916365140796, 1.35545694658076012940219470133, 2.63733332337251842759318066888, 4.77502228234619983121960694797, 5.81961700527345536521701207216, 6.55591996534890668364474488864, 8.319796317392223304207199935505, 9.127008183001210530996366308169, 10.00584761192644607216704929770, 11.17782204643352514113980634081, 11.37682255114319400341316314292

Graph of the ZZ-function along the critical line