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Properties

Label 2-14e2-196.111-c1-0-22
Degree 22
Conductor 196196
Sign −0.676+0.736i-0.676 + 0.736i
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  + (0.145 − 1.40i)2-s + (−0.0665 − 0.0834i)3-s + (−1.95 − 0.408i)4-s + (1.11 − 0.890i)5-s + (−0.127 + 0.0814i)6-s + (−1.34 − 2.27i)7-s + (−0.859 + 2.69i)8-s + (0.665 − 2.91i)9-s + (−1.09 − 1.70i)10-s + (−1.57 + 0.359i)11-s + (0.0961 + 0.190i)12-s + (0.776 − 0.177i)13-s + (−3.40 + 1.55i)14-s + (−0.148 − 0.0339i)15-s + (3.66 + 1.60i)16-s + (1.17 − 2.44i)17-s + ⋯
L(s)  = 1  + (0.102 − 0.994i)2-s + (−0.0384 − 0.0481i)3-s + (−0.978 − 0.204i)4-s + (0.499 − 0.398i)5-s + (−0.0518 + 0.0332i)6-s + (−0.507 − 0.861i)7-s + (−0.303 + 0.952i)8-s + (0.221 − 0.971i)9-s + (−0.344 − 0.537i)10-s + (−0.474 + 0.108i)11-s + (0.0277 + 0.0549i)12-s + (0.215 − 0.0491i)13-s + (−0.909 + 0.416i)14-s + (−0.0383 − 0.00875i)15-s + (0.916 + 0.400i)16-s + (0.285 − 0.593i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((−0.676+0.736i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((−0.676+0.736i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −0.676+0.736i-0.676 + 0.736i
Analytic conductor: 1.565061.56506
Root analytic conductor: 1.251021.25102
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ196(111,⋅)\chi_{196} (111, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 196, ( :1/2), −0.676+0.736i)(2,\ 196,\ (\ :1/2),\ -0.676 + 0.736i)

Particular Values

L(1)L(1) ≈\approx 0.442063−1.00639i0.442063 - 1.00639i
L(12)L(\frac12) ≈\approx 0.442063−1.00639i0.442063 - 1.00639i
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+(−0.145+1.40i)T 1 + (-0.145 + 1.40i)T
7 1+(1.34+2.27i)T 1 + (1.34 + 2.27i)T
good3 1+(0.0665+0.0834i)T+(−0.667+2.92i)T2 1 + (0.0665 + 0.0834i)T + (-0.667 + 2.92i)T^{2}
5 1+(−1.11+0.890i)T+(1.11−4.87i)T2 1 + (-1.11 + 0.890i)T + (1.11 - 4.87i)T^{2}
11 1+(1.57−0.359i)T+(9.91−4.77i)T2 1 + (1.57 - 0.359i)T + (9.91 - 4.77i)T^{2}
13 1+(−0.776+0.177i)T+(11.7−5.64i)T2 1 + (-0.776 + 0.177i)T + (11.7 - 5.64i)T^{2}
17 1+(−1.17+2.44i)T+(−10.5−13.2i)T2 1 + (-1.17 + 2.44i)T + (-10.5 - 13.2i)T^{2}
19 1+0.719T+19T2 1 + 0.719T + 19T^{2}
23 1+(−1.19−2.49i)T+(−14.3+17.9i)T2 1 + (-1.19 - 2.49i)T + (-14.3 + 17.9i)T^{2}
29 1+(−3.80−1.83i)T+(18.0+22.6i)T2 1 + (-3.80 - 1.83i)T + (18.0 + 22.6i)T^{2}
31 1−7.03T+31T2 1 - 7.03T + 31T^{2}
37 1+(−4.28−2.06i)T+(23.0+28.9i)T2 1 + (-4.28 - 2.06i)T + (23.0 + 28.9i)T^{2}
41 1+(0.875−0.697i)T+(9.12−39.9i)T2 1 + (0.875 - 0.697i)T + (9.12 - 39.9i)T^{2}
43 1+(−4.71−3.75i)T+(9.56+41.9i)T2 1 + (-4.71 - 3.75i)T + (9.56 + 41.9i)T^{2}
47 1+(2.47+10.8i)T+(−42.3+20.3i)T2 1 + (2.47 + 10.8i)T + (-42.3 + 20.3i)T^{2}
53 1+(−2.38+1.14i)T+(33.0−41.4i)T2 1 + (-2.38 + 1.14i)T + (33.0 - 41.4i)T^{2}
59 1+(−8.49+10.6i)T+(−13.1−57.5i)T2 1 + (-8.49 + 10.6i)T + (-13.1 - 57.5i)T^{2}
61 1+(4.31−8.95i)T+(−38.0−47.6i)T2 1 + (4.31 - 8.95i)T + (-38.0 - 47.6i)T^{2}
67 1−2.88iT−67T2 1 - 2.88iT - 67T^{2}
71 1+(−5.12−10.6i)T+(−44.2+55.5i)T2 1 + (-5.12 - 10.6i)T + (-44.2 + 55.5i)T^{2}
73 1+(12.1+2.76i)T+(65.7+31.6i)T2 1 + (12.1 + 2.76i)T + (65.7 + 31.6i)T^{2}
79 1−9.74iT−79T2 1 - 9.74iT - 79T^{2}
83 1+(1.87−8.19i)T+(−74.7−36.0i)T2 1 + (1.87 - 8.19i)T + (-74.7 - 36.0i)T^{2}
89 1+(14.2+3.26i)T+(80.1+38.6i)T2 1 + (14.2 + 3.26i)T + (80.1 + 38.6i)T^{2}
97 1+1.72iT−97T2 1 + 1.72iT - 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

−12.19310196199863837583436697851, −11.20681652885477815320466327789, −10.00412580571617084446762336462, −9.615646575768828195631721442712, −8.400046244856297833161255978013, −6.91198650816422109500897980087, −5.55690848780836364230990286836, −4.26065452701599219306413351151, −3.03215516838233865046455311716, −1.04710076390748662971430303562, 2.69533899470714895755467944584, 4.51801021622617407363719815332, 5.74466320475959903705797983310, 6.45637380444187991729280989899, 7.80242406009617189118765897568, 8.655491683964041868713238895496, 9.831899126393925475244722937134, 10.61488924170481838937681369093, 12.19551965102831570196847467793, 13.09592876150736419379387827907

Graph of the ZZ-function along the critical line