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Properties

Label 2-14e2-196.103-c1-0-21
Degree 22
Conductor 196196
Sign −0.974+0.224i-0.974 + 0.224i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.19 − 0.748i)2-s + (0.318 + 0.217i)3-s + (0.879 + 1.79i)4-s + (−2.23 − 0.167i)5-s + (−0.219 − 0.499i)6-s + (−2.62 + 0.318i)7-s + (0.289 − 2.81i)8-s + (−1.04 − 2.65i)9-s + (2.55 + 1.87i)10-s + (−4.62 − 1.81i)11-s + (−0.110 + 0.763i)12-s + (4.05 − 3.23i)13-s + (3.38 + 1.58i)14-s + (−0.675 − 0.538i)15-s + (−2.45 + 3.15i)16-s + (−1.83 − 1.97i)17-s + ⋯
L(s)  = 1  + (−0.848 − 0.529i)2-s + (0.184 + 0.125i)3-s + (0.439 + 0.898i)4-s + (−0.998 − 0.0748i)5-s + (−0.0897 − 0.203i)6-s + (−0.992 + 0.120i)7-s + (0.102 − 0.994i)8-s + (−0.347 − 0.884i)9-s + (0.807 + 0.591i)10-s + (−1.39 − 0.547i)11-s + (−0.0317 + 0.220i)12-s + (1.12 − 0.897i)13-s + (0.905 + 0.423i)14-s + (−0.174 − 0.139i)15-s + (−0.613 + 0.789i)16-s + (−0.445 − 0.480i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((−0.974+0.224i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((−0.974+0.224i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −0.974+0.224i-0.974 + 0.224i
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), −0.974+0.224i)(2,\ 196,\ (\ :1/2),\ -0.974 + 0.224i)

Particular Values

L(1)L(1) ≈\approx 0.0280597−0.246733i0.0280597 - 0.246733i
L(12)L(\frac12) ≈\approx 0.0280597−0.246733i0.0280597 - 0.246733i
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+(1.19+0.748i)T 1 + (1.19 + 0.748i)T
7 1+(2.62−0.318i)T 1 + (2.62 - 0.318i)T
good3 1+(−0.318−0.217i)T+(1.09+2.79i)T2 1 + (-0.318 - 0.217i)T + (1.09 + 2.79i)T^{2}
5 1+(2.23+0.167i)T+(4.94+0.745i)T2 1 + (2.23 + 0.167i)T + (4.94 + 0.745i)T^{2}
11 1+(4.62+1.81i)T+(8.06+7.48i)T2 1 + (4.62 + 1.81i)T + (8.06 + 7.48i)T^{2}
13 1+(−4.05+3.23i)T+(2.89−12.6i)T2 1 + (-4.05 + 3.23i)T + (2.89 - 12.6i)T^{2}
17 1+(1.83+1.97i)T+(−1.27+16.9i)T2 1 + (1.83 + 1.97i)T + (-1.27 + 16.9i)T^{2}
19 1+(1.20−2.08i)T+(−9.5−16.4i)T2 1 + (1.20 - 2.08i)T + (-9.5 - 16.4i)T^{2}
23 1+(2.13−2.29i)T+(−1.71−22.9i)T2 1 + (2.13 - 2.29i)T + (-1.71 - 22.9i)T^{2}
29 1+(−1.93−8.48i)T+(−26.1+12.5i)T2 1 + (-1.93 - 8.48i)T + (-26.1 + 12.5i)T^{2}
31 1+(0.0585+0.101i)T+(−15.5+26.8i)T2 1 + (0.0585 + 0.101i)T + (-15.5 + 26.8i)T^{2}
37 1+(0.160+0.0494i)T+(30.5+20.8i)T2 1 + (0.160 + 0.0494i)T + (30.5 + 20.8i)T^{2}
41 1+(4.98+10.3i)T+(−25.5+32.0i)T2 1 + (4.98 + 10.3i)T + (-25.5 + 32.0i)T^{2}
43 1+(1.97−4.09i)T+(−26.8−33.6i)T2 1 + (1.97 - 4.09i)T + (-26.8 - 33.6i)T^{2}
47 1+(−10.2+1.54i)T+(44.9−13.8i)T2 1 + (-10.2 + 1.54i)T + (44.9 - 13.8i)T^{2}
53 1+(−2.75+0.850i)T+(43.7−29.8i)T2 1 + (-2.75 + 0.850i)T + (43.7 - 29.8i)T^{2}
59 1+(0.758+10.1i)T+(−58.3+8.79i)T2 1 + (0.758 + 10.1i)T + (-58.3 + 8.79i)T^{2}
61 1+(2.62−8.51i)T+(−50.4−34.3i)T2 1 + (2.62 - 8.51i)T + (-50.4 - 34.3i)T^{2}
67 1+(−5.07+2.92i)T+(33.5−58.0i)T2 1 + (-5.07 + 2.92i)T + (33.5 - 58.0i)T^{2}
71 1+(9.49+2.16i)T+(63.9+30.8i)T2 1 + (9.49 + 2.16i)T + (63.9 + 30.8i)T^{2}
73 1+(−1.11+7.37i)T+(−69.7−21.5i)T2 1 + (-1.11 + 7.37i)T + (-69.7 - 21.5i)T^{2}
79 1+(11.9+6.92i)T+(39.5+68.4i)T2 1 + (11.9 + 6.92i)T + (39.5 + 68.4i)T^{2}
83 1+(1.95−2.45i)T+(−18.4−80.9i)T2 1 + (1.95 - 2.45i)T + (-18.4 - 80.9i)T^{2}
89 1+(1.74−0.684i)T+(65.2−60.5i)T2 1 + (1.74 - 0.684i)T + (65.2 - 60.5i)T^{2}
97 1+1.56iT−97T2 1 + 1.56iT - 97T^{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

−11.98424331517771981475810225820, −10.90983694221218018105723073487, −10.18571028739531629491172262597, −8.904705948490811522484078671641, −8.307778144503861947582871922615, −7.21072890055404698518126163787, −5.87363996315331550165698967745, −3.70093868573609931711072338019, −3.03280205363998831721849229211, −0.26228651585725270087900439901, 2.47211402794670663352728996239, 4.35845186989736974727239908425, 5.95007595541657736395751627833, 7.04802250766181684490685186030, 7.992224809273150792254701939019, 8.676685092625751106950127029643, 10.00732704776201795494493680119, 10.80807618357789610867339069991, 11.70496125383986532430676720573, 13.12769080525656103484433998229

Graph of the ZZ-function along the critical line