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Properties

Label 2-14e2-196.103-c1-0-8
Degree 22
Conductor 196196
Sign 0.996+0.0853i0.996 + 0.0853i
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  + (−0.925 + 1.06i)2-s + (−0.318 − 0.217i)3-s + (−0.285 − 1.97i)4-s + (−2.23 − 0.167i)5-s + (0.527 − 0.139i)6-s + (2.62 − 0.318i)7-s + (2.38 + 1.52i)8-s + (−1.04 − 2.65i)9-s + (2.24 − 2.23i)10-s + (4.62 + 1.81i)11-s + (−0.339 + 0.692i)12-s + (4.05 − 3.23i)13-s + (−2.09 + 3.10i)14-s + (0.675 + 0.538i)15-s + (−3.83 + 1.13i)16-s + (−1.83 − 1.97i)17-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)2-s + (−0.184 − 0.125i)3-s + (−0.142 − 0.989i)4-s + (−0.998 − 0.0748i)5-s + (0.215 − 0.0569i)6-s + (0.992 − 0.120i)7-s + (0.841 + 0.540i)8-s + (−0.347 − 0.884i)9-s + (0.710 − 0.705i)10-s + (1.39 + 0.547i)11-s + (−0.0979 + 0.200i)12-s + (1.12 − 0.897i)13-s + (−0.558 + 0.829i)14-s + (0.174 + 0.139i)15-s + (−0.959 + 0.282i)16-s + (−0.445 − 0.480i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((0.996+0.0853i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0853i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((0.996+0.0853i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0853i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: 0.996+0.0853i0.996 + 0.0853i
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), 0.996+0.0853i)(2,\ 196,\ (\ :1/2),\ 0.996 + 0.0853i)

Particular Values

L(1)L(1) ≈\approx 0.783609−0.0334839i0.783609 - 0.0334839i
L(12)L(\frac12) ≈\approx 0.783609−0.0334839i0.783609 - 0.0334839i
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+(0.925−1.06i)T 1 + (0.925 - 1.06i)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

−12.18580905845330917126342474402, −11.43128391697142906483020516600, −10.61352311619570416411641059421, −9.040082233303129324680352847195, −8.560739304282329393554353659131, −7.35600127166260757109555536761, −6.55236938268697636778500257895, −5.16043563273760311897836698210, −3.86890851423148176214128344192, −1.05974346359626297485786138455, 1.62341411520913351249910895030, 3.63359764209158424739184669782, 4.54203956821445050933237764688, 6.44255305665694440388119055116, 7.989433049087123970633602604150, 8.362165584010205082204063422297, 9.522327287036003405498955572962, 11.04051669843623766861985653371, 11.38332573142840841394170798115, 11.88173138296845707128044000020

Graph of the ZZ-function along the critical line