↑

Properties

Label 2-308-77.58-c1-0-1
Degree 22
Conductor 308308
Sign −0.276−0.960i-0.276 - 0.960i
Analytic cond. 2.459392.45939
Root an. cond. 1.568241.56824
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  + (0.599 − 0.665i)3-s + (−4.03 + 1.79i)5-s + (0.860 + 2.50i)7-s + (0.229 + 2.18i)9-s + (−0.394 − 3.29i)11-s + (−3.36 + 2.44i)13-s + (−1.22 + 3.76i)15-s + (−0.518 + 4.93i)17-s + (−0.672 − 0.142i)19-s + (2.18 + 0.927i)21-s + (−2.70 + 4.67i)23-s + (9.69 − 10.7i)25-s + (3.76 + 2.73i)27-s + (1.02 − 3.15i)29-s + (6.03 + 2.68i)31-s + ⋯
L(s)  = 1  + (0.346 − 0.384i)3-s + (−1.80 + 0.802i)5-s + (0.325 + 0.945i)7-s + (0.0765 + 0.728i)9-s + (−0.119 − 0.992i)11-s + (−0.933 + 0.678i)13-s + (−0.315 + 0.970i)15-s + (−0.125 + 1.19i)17-s + (−0.154 − 0.0327i)19-s + (0.475 + 0.202i)21-s + (−0.563 + 0.975i)23-s + (1.93 − 2.15i)25-s + (0.724 + 0.526i)27-s + (0.190 − 0.586i)29-s + (1.08 + 0.482i)31-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=((−0.276−0.960i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+1/2) L(s)=((−0.276−0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: −0.276−0.960i-0.276 - 0.960i
Analytic conductor: 2.459392.45939
Root analytic conductor: 1.568241.56824
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ308(289,⋅)\chi_{308} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :1/2), −0.276−0.960i)(2,\ 308,\ (\ :1/2),\ -0.276 - 0.960i)

Particular Values

L(1)L(1) ≈\approx 0.482827+0.641652i0.482827 + 0.641652i
L(12)L(\frac12) ≈\approx 0.482827+0.641652i0.482827 + 0.641652i
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)
bad2 1 1
7 1+(−0.860−2.50i)T 1 + (-0.860 - 2.50i)T
11 1+(0.394+3.29i)T 1 + (0.394 + 3.29i)T
good3 1+(−0.599+0.665i)T+(−0.313−2.98i)T2 1 + (-0.599 + 0.665i)T + (-0.313 - 2.98i)T^{2}
5 1+(4.03−1.79i)T+(3.34−3.71i)T2 1 + (4.03 - 1.79i)T + (3.34 - 3.71i)T^{2}
13 1+(3.36−2.44i)T+(4.01−12.3i)T2 1 + (3.36 - 2.44i)T + (4.01 - 12.3i)T^{2}
17 1+(0.518−4.93i)T+(−16.6−3.53i)T2 1 + (0.518 - 4.93i)T + (-16.6 - 3.53i)T^{2}
19 1+(0.672+0.142i)T+(17.3+7.72i)T2 1 + (0.672 + 0.142i)T + (17.3 + 7.72i)T^{2}
23 1+(2.70−4.67i)T+(−11.5−19.9i)T2 1 + (2.70 - 4.67i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.02+3.15i)T+(−23.4−17.0i)T2 1 + (-1.02 + 3.15i)T + (-23.4 - 17.0i)T^{2}
31 1+(−6.03−2.68i)T+(20.7+23.0i)T2 1 + (-6.03 - 2.68i)T + (20.7 + 23.0i)T^{2}
37 1+(1.71+1.90i)T+(−3.86+36.7i)T2 1 + (1.71 + 1.90i)T + (-3.86 + 36.7i)T^{2}
41 1+(1.30+4.02i)T+(−33.1+24.0i)T2 1 + (1.30 + 4.02i)T + (-33.1 + 24.0i)T^{2}
43 1−4.76T+43T2 1 - 4.76T + 43T^{2}
47 1+(3.55+0.754i)T+(42.9+19.1i)T2 1 + (3.55 + 0.754i)T + (42.9 + 19.1i)T^{2}
53 1+(1.29+0.577i)T+(35.4+39.3i)T2 1 + (1.29 + 0.577i)T + (35.4 + 39.3i)T^{2}
59 1+(5.60−1.19i)T+(53.8−23.9i)T2 1 + (5.60 - 1.19i)T + (53.8 - 23.9i)T^{2}
61 1+(−5.37+2.39i)T+(40.8−45.3i)T2 1 + (-5.37 + 2.39i)T + (40.8 - 45.3i)T^{2}
67 1+(−0.225−0.390i)T+(−33.5+58.0i)T2 1 + (-0.225 - 0.390i)T + (-33.5 + 58.0i)T^{2}
71 1+(7.43+5.39i)T+(21.9+67.5i)T2 1 + (7.43 + 5.39i)T + (21.9 + 67.5i)T^{2}
73 1+(−2.52+0.535i)T+(66.6−29.6i)T2 1 + (-2.52 + 0.535i)T + (66.6 - 29.6i)T^{2}
79 1+(−1.14−10.8i)T+(−77.2+16.4i)T2 1 + (-1.14 - 10.8i)T + (-77.2 + 16.4i)T^{2}
83 1+(−5.68−4.13i)T+(25.6+78.9i)T2 1 + (-5.68 - 4.13i)T + (25.6 + 78.9i)T^{2}
89 1+(−1.82+3.16i)T+(−44.5−77.0i)T2 1 + (-1.82 + 3.16i)T + (-44.5 - 77.0i)T^{2}
97 1+(−4.83+3.51i)T+(29.9−92.2i)T2 1 + (-4.83 + 3.51i)T + (29.9 - 92.2i)T^{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

−11.85545303151644315525165039742, −11.25193644532896253934737983728, −10.36063944301749340500539060264, −8.730971394445857228624319266979, −8.083468820455602458220564159755, −7.43830233730852169366798547590, −6.26416741328062013466936521249, −4.75548752783105289121966232858, −3.53906823720121875625595734869, −2.34379219165138890184372935602, 0.56232238394200371749727788912, 3.16364014185718523159343262840, 4.44506133116627794108843203967, 4.69814580174613838770864530166, 6.94535814112030108345131760111, 7.65687017861867766200764831653, 8.433092894201859486241735598526, 9.544779404324327557017515294878, 10.43259742683773591774922521570, 11.64737010147357683969736685516

Graph of the ZZ-function along the critical line