↑

Properties

Label 2-14e2-196.103-c1-0-24
Degree 22
Conductor 196196
Sign −0.154−0.987i-0.154 - 0.987i
Analytic cond. 1.565061.56506
Root an. cond. 1.251021.25102
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  + (−0.167 − 1.40i)2-s + (−1.97 − 1.34i)3-s + (−1.94 + 0.471i)4-s + (−0.0159 − 0.00119i)5-s + (−1.56 + 3.00i)6-s + (−2.59 + 0.509i)7-s + (0.987 + 2.65i)8-s + (0.999 + 2.54i)9-s + (0.000997 + 0.0225i)10-s + (1.65 + 0.649i)11-s + (4.48 + 1.69i)12-s + (−2.07 + 1.65i)13-s + (1.15 + 3.56i)14-s + (0.0299 + 0.0238i)15-s + (3.55 − 1.83i)16-s + (−3.29 − 3.55i)17-s + ⋯
L(s)  = 1  + (−0.118 − 0.992i)2-s + (−1.14 − 0.778i)3-s + (−0.971 + 0.235i)4-s + (−0.00713 − 0.000534i)5-s + (−0.637 + 1.22i)6-s + (−0.981 + 0.192i)7-s + (0.349 + 0.937i)8-s + (0.333 + 0.849i)9-s + (0.000315 + 0.00714i)10-s + (0.499 + 0.195i)11-s + (1.29 + 0.487i)12-s + (−0.574 + 0.458i)13-s + (0.307 + 0.951i)14-s + (0.00773 + 0.00616i)15-s + (0.888 − 0.457i)16-s + (−0.799 − 0.861i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((−0.154−0.987i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((−0.154−0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −0.154−0.987i-0.154 - 0.987i
Analytic conductor: 1.565061.56506
Root analytic conductor: 1.251021.25102
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ196(103,⋅)\chi_{196} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 196, ( :1/2), −0.154−0.987i)(2,\ 196,\ (\ :1/2),\ -0.154 - 0.987i)

Particular Values

L(1)L(1) ≈\approx 0.0512515+0.0598973i0.0512515 + 0.0598973i
L(12)L(\frac12) ≈\approx 0.0512515+0.0598973i0.0512515 + 0.0598973i
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+(0.167+1.40i)T 1 + (0.167 + 1.40i)T
7 1+(2.59−0.509i)T 1 + (2.59 - 0.509i)T
good3 1+(1.97+1.34i)T+(1.09+2.79i)T2 1 + (1.97 + 1.34i)T + (1.09 + 2.79i)T^{2}
5 1+(0.0159+0.00119i)T+(4.94+0.745i)T2 1 + (0.0159 + 0.00119i)T + (4.94 + 0.745i)T^{2}
11 1+(−1.65−0.649i)T+(8.06+7.48i)T2 1 + (-1.65 - 0.649i)T + (8.06 + 7.48i)T^{2}
13 1+(2.07−1.65i)T+(2.89−12.6i)T2 1 + (2.07 - 1.65i)T + (2.89 - 12.6i)T^{2}
17 1+(3.29+3.55i)T+(−1.27+16.9i)T2 1 + (3.29 + 3.55i)T + (-1.27 + 16.9i)T^{2}
19 1+(0.682−1.18i)T+(−9.5−16.4i)T2 1 + (0.682 - 1.18i)T + (-9.5 - 16.4i)T^{2}
23 1+(3.75−4.04i)T+(−1.71−22.9i)T2 1 + (3.75 - 4.04i)T + (-1.71 - 22.9i)T^{2}
29 1+(−0.487−2.13i)T+(−26.1+12.5i)T2 1 + (-0.487 - 2.13i)T + (-26.1 + 12.5i)T^{2}
31 1+(4.96+8.59i)T+(−15.5+26.8i)T2 1 + (4.96 + 8.59i)T + (-15.5 + 26.8i)T^{2}
37 1+(−1.78−0.549i)T+(30.5+20.8i)T2 1 + (-1.78 - 0.549i)T + (30.5 + 20.8i)T^{2}
41 1+(1.78+3.71i)T+(−25.5+32.0i)T2 1 + (1.78 + 3.71i)T + (-25.5 + 32.0i)T^{2}
43 1+(−4.01+8.34i)T+(−26.8−33.6i)T2 1 + (-4.01 + 8.34i)T + (-26.8 - 33.6i)T^{2}
47 1+(9.62−1.45i)T+(44.9−13.8i)T2 1 + (9.62 - 1.45i)T + (44.9 - 13.8i)T^{2}
53 1+(8.11−2.50i)T+(43.7−29.8i)T2 1 + (8.11 - 2.50i)T + (43.7 - 29.8i)T^{2}
59 1+(−0.462−6.17i)T+(−58.3+8.79i)T2 1 + (-0.462 - 6.17i)T + (-58.3 + 8.79i)T^{2}
61 1+(−2.72+8.81i)T+(−50.4−34.3i)T2 1 + (-2.72 + 8.81i)T + (-50.4 - 34.3i)T^{2}
67 1+(−4.78+2.76i)T+(33.5−58.0i)T2 1 + (-4.78 + 2.76i)T + (33.5 - 58.0i)T^{2}
71 1+(−13.9−3.18i)T+(63.9+30.8i)T2 1 + (-13.9 - 3.18i)T + (63.9 + 30.8i)T^{2}
73 1+(1.28−8.51i)T+(−69.7−21.5i)T2 1 + (1.28 - 8.51i)T + (-69.7 - 21.5i)T^{2}
79 1+(0.651+0.375i)T+(39.5+68.4i)T2 1 + (0.651 + 0.375i)T + (39.5 + 68.4i)T^{2}
83 1+(−2.16+2.71i)T+(−18.4−80.9i)T2 1 + (-2.16 + 2.71i)T + (-18.4 - 80.9i)T^{2}
89 1+(−1.89+0.744i)T+(65.2−60.5i)T2 1 + (-1.89 + 0.744i)T + (65.2 - 60.5i)T^{2}
97 1+12.1iT−97T2 1 + 12.1iT - 97T^{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

−11.78968750499295888746913204706, −11.19762458453048150160898567400, −9.857678159359645039657577270513, −9.215508342576611526518926131722, −7.57627110915766591825904086813, −6.47952129123934021518746756745, −5.38541222174242879241800589554, −3.89203718517378298923582742864, −2.07587862909019989010759496339, −0.079650395628756524293527427063, 3.83655563345321916858319296925, 4.89340595710004538381780185396, 6.07857751754432201688332812471, 6.64211749510933408838798826721, 8.129401740114455812258760628264, 9.420628947413872603865293033848, 10.12015065902249145816278007160, 11.01548159421197706366578980666, 12.35102001220771188112962975248, 13.15924787150462195394165345513

Graph of the ZZ-function along the critical line