↑

Properties

Label 2-245-245.157-c1-0-13
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

Related objects

Downloads

Learn more

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(157,⋅)\chi_{245} (157, \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}
show more
show less
   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

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

Graph of the ZZ-function along the critical line