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Properties

Label 2-14e2-196.103-c1-0-22
Degree 22
Conductor 196196
Sign −0.430+0.902i-0.430 + 0.902i
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  + (0.852 − 1.12i)2-s + (−0.330 − 0.225i)3-s + (−0.547 − 1.92i)4-s + (−0.194 − 0.0145i)5-s + (−0.535 + 0.180i)6-s + (−1.62 − 2.09i)7-s + (−2.63 − 1.02i)8-s + (−1.03 − 2.64i)9-s + (−0.181 + 0.206i)10-s + (4.10 + 1.60i)11-s + (−0.252 + 0.758i)12-s + (2.89 − 2.31i)13-s + (−3.74 + 0.0466i)14-s + (0.0608 + 0.0485i)15-s + (−3.39 + 2.10i)16-s + (4.17 + 4.50i)17-s + ⋯
L(s)  = 1  + (0.602 − 0.798i)2-s + (−0.190 − 0.130i)3-s + (−0.273 − 0.961i)4-s + (−0.0868 − 0.00650i)5-s + (−0.218 + 0.0738i)6-s + (−0.612 − 0.790i)7-s + (−0.932 − 0.360i)8-s + (−0.345 − 0.881i)9-s + (−0.0575 + 0.0653i)10-s + (1.23 + 0.485i)11-s + (−0.0727 + 0.219i)12-s + (0.804 − 0.641i)13-s + (−0.999 + 0.0124i)14-s + (0.0157 + 0.0125i)15-s + (−0.849 + 0.526i)16-s + (1.01 + 1.09i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((−0.430+0.902i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((−0.430+0.902i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −0.430+0.902i-0.430 + 0.902i
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), −0.430+0.902i)(2,\ 196,\ (\ :1/2),\ -0.430 + 0.902i)

Particular Values

L(1)L(1) ≈\approx 0.730655−1.15822i0.730655 - 1.15822i
L(12)L(\frac12) ≈\approx 0.730655−1.15822i0.730655 - 1.15822i
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+(−0.852+1.12i)T 1 + (-0.852 + 1.12i)T
7 1+(1.62+2.09i)T 1 + (1.62 + 2.09i)T
good3 1+(0.330+0.225i)T+(1.09+2.79i)T2 1 + (0.330 + 0.225i)T + (1.09 + 2.79i)T^{2}
5 1+(0.194+0.0145i)T+(4.94+0.745i)T2 1 + (0.194 + 0.0145i)T + (4.94 + 0.745i)T^{2}
11 1+(−4.10−1.60i)T+(8.06+7.48i)T2 1 + (-4.10 - 1.60i)T + (8.06 + 7.48i)T^{2}
13 1+(−2.89+2.31i)T+(2.89−12.6i)T2 1 + (-2.89 + 2.31i)T + (2.89 - 12.6i)T^{2}
17 1+(−4.17−4.50i)T+(−1.27+16.9i)T2 1 + (-4.17 - 4.50i)T + (-1.27 + 16.9i)T^{2}
19 1+(2.69−4.66i)T+(−9.5−16.4i)T2 1 + (2.69 - 4.66i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.58+2.79i)T+(−1.71−22.9i)T2 1 + (-2.58 + 2.79i)T + (-1.71 - 22.9i)T^{2}
29 1+(1.16+5.11i)T+(−26.1+12.5i)T2 1 + (1.16 + 5.11i)T + (-26.1 + 12.5i)T^{2}
31 1+(−4.63−8.03i)T+(−15.5+26.8i)T2 1 + (-4.63 - 8.03i)T + (-15.5 + 26.8i)T^{2}
37 1+(−10.0−3.09i)T+(30.5+20.8i)T2 1 + (-10.0 - 3.09i)T + (30.5 + 20.8i)T^{2}
41 1+(1.28+2.66i)T+(−25.5+32.0i)T2 1 + (1.28 + 2.66i)T + (-25.5 + 32.0i)T^{2}
43 1+(−0.871+1.81i)T+(−26.8−33.6i)T2 1 + (-0.871 + 1.81i)T + (-26.8 - 33.6i)T^{2}
47 1+(−0.800+0.120i)T+(44.9−13.8i)T2 1 + (-0.800 + 0.120i)T + (44.9 - 13.8i)T^{2}
53 1+(5.82−1.79i)T+(43.7−29.8i)T2 1 + (5.82 - 1.79i)T + (43.7 - 29.8i)T^{2}
59 1+(0.925+12.3i)T+(−58.3+8.79i)T2 1 + (0.925 + 12.3i)T + (-58.3 + 8.79i)T^{2}
61 1+(−2.27+7.37i)T+(−50.4−34.3i)T2 1 + (-2.27 + 7.37i)T + (-50.4 - 34.3i)T^{2}
67 1+(−1.40+0.808i)T+(33.5−58.0i)T2 1 + (-1.40 + 0.808i)T + (33.5 - 58.0i)T^{2}
71 1+(4.57+1.04i)T+(63.9+30.8i)T2 1 + (4.57 + 1.04i)T + (63.9 + 30.8i)T^{2}
73 1+(2.28−15.1i)T+(−69.7−21.5i)T2 1 + (2.28 - 15.1i)T + (-69.7 - 21.5i)T^{2}
79 1+(2.58+1.49i)T+(39.5+68.4i)T2 1 + (2.58 + 1.49i)T + (39.5 + 68.4i)T^{2}
83 1+(0.292−0.366i)T+(−18.4−80.9i)T2 1 + (0.292 - 0.366i)T + (-18.4 - 80.9i)T^{2}
89 1+(10.1−3.99i)T+(65.2−60.5i)T2 1 + (10.1 - 3.99i)T + (65.2 - 60.5i)T^{2}
97 1−4.71iT−97T2 1 - 4.71iT - 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

−12.35151997198541639526093115626, −11.35739981024985496122271107854, −10.28899391599519325358415659328, −9.611175415252447571461743411089, −8.289403903156489036760589665777, −6.52167946866640287989799073810, −5.98479336535887875882400897261, −4.11574656073446555889938222223, −3.43273824148722965156112539099, −1.18957755622566829878345638954, 2.90583638771697462693997494596, 4.28653408680468697590565525792, 5.60920202206980605029643631210, 6.36615866425303966821139653577, 7.57816484499857025304990620835, 8.808243396499466495328308242265, 9.465662537645349459272174442348, 11.41953092538784297820965247552, 11.69020072207595928552992320390, 13.09598554744989082201461879252

Graph of the ZZ-function along the critical line