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Properties

Label 2-14e2-196.103-c1-0-3
Degree 22
Conductor 196196
Sign −0.186−0.982i-0.186 - 0.982i
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  + (−1.37 + 0.340i)2-s + (2.65 + 1.81i)3-s + (1.76 − 0.934i)4-s + (−3.39 − 0.254i)5-s + (−4.26 − 1.58i)6-s + (0.227 + 2.63i)7-s + (−2.10 + 1.88i)8-s + (2.68 + 6.83i)9-s + (4.74 − 0.806i)10-s + (−0.627 − 0.246i)11-s + (6.39 + 0.719i)12-s + (0.189 − 0.151i)13-s + (−1.20 − 3.54i)14-s + (−8.56 − 6.82i)15-s + (2.25 − 3.30i)16-s + (2.64 + 2.84i)17-s + ⋯
L(s)  = 1  + (−0.970 + 0.240i)2-s + (1.53 + 1.04i)3-s + (0.884 − 0.467i)4-s + (−1.51 − 0.113i)5-s + (−1.74 − 0.645i)6-s + (0.0858 + 0.996i)7-s + (−0.745 + 0.666i)8-s + (0.894 + 2.27i)9-s + (1.50 − 0.255i)10-s + (−0.189 − 0.0742i)11-s + (1.84 + 0.207i)12-s + (0.0525 − 0.0419i)13-s + (−0.323 − 0.946i)14-s + (−2.21 − 1.76i)15-s + (0.563 − 0.826i)16-s + (0.641 + 0.691i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((−0.186−0.982i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((−0.186−0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −0.186−0.982i-0.186 - 0.982i
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), −0.186−0.982i)(2,\ 196,\ (\ :1/2),\ -0.186 - 0.982i)

Particular Values

L(1)L(1) ≈\approx 0.641027+0.774475i0.641027 + 0.774475i
L(12)L(\frac12) ≈\approx 0.641027+0.774475i0.641027 + 0.774475i
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+(1.37−0.340i)T 1 + (1.37 - 0.340i)T
7 1+(−0.227−2.63i)T 1 + (-0.227 - 2.63i)T
good3 1+(−2.65−1.81i)T+(1.09+2.79i)T2 1 + (-2.65 - 1.81i)T + (1.09 + 2.79i)T^{2}
5 1+(3.39+0.254i)T+(4.94+0.745i)T2 1 + (3.39 + 0.254i)T + (4.94 + 0.745i)T^{2}
11 1+(0.627+0.246i)T+(8.06+7.48i)T2 1 + (0.627 + 0.246i)T + (8.06 + 7.48i)T^{2}
13 1+(−0.189+0.151i)T+(2.89−12.6i)T2 1 + (-0.189 + 0.151i)T + (2.89 - 12.6i)T^{2}
17 1+(−2.64−2.84i)T+(−1.27+16.9i)T2 1 + (-2.64 - 2.84i)T + (-1.27 + 16.9i)T^{2}
19 1+(−1.77+3.07i)T+(−9.5−16.4i)T2 1 + (-1.77 + 3.07i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.03+3.27i)T+(−1.71−22.9i)T2 1 + (-3.03 + 3.27i)T + (-1.71 - 22.9i)T^{2}
29 1+(0.473+2.07i)T+(−26.1+12.5i)T2 1 + (0.473 + 2.07i)T + (-26.1 + 12.5i)T^{2}
31 1+(−0.307−0.532i)T+(−15.5+26.8i)T2 1 + (-0.307 - 0.532i)T + (-15.5 + 26.8i)T^{2}
37 1+(1.57+0.485i)T+(30.5+20.8i)T2 1 + (1.57 + 0.485i)T + (30.5 + 20.8i)T^{2}
41 1+(1.74+3.61i)T+(−25.5+32.0i)T2 1 + (1.74 + 3.61i)T + (-25.5 + 32.0i)T^{2}
43 1+(−2.65+5.50i)T+(−26.8−33.6i)T2 1 + (-2.65 + 5.50i)T + (-26.8 - 33.6i)T^{2}
47 1+(−0.966+0.145i)T+(44.9−13.8i)T2 1 + (-0.966 + 0.145i)T + (44.9 - 13.8i)T^{2}
53 1+(−8.02+2.47i)T+(43.7−29.8i)T2 1 + (-8.02 + 2.47i)T + (43.7 - 29.8i)T^{2}
59 1+(0.668+8.92i)T+(−58.3+8.79i)T2 1 + (0.668 + 8.92i)T + (-58.3 + 8.79i)T^{2}
61 1+(1.03−3.34i)T+(−50.4−34.3i)T2 1 + (1.03 - 3.34i)T + (-50.4 - 34.3i)T^{2}
67 1+(−10.4+6.03i)T+(33.5−58.0i)T2 1 + (-10.4 + 6.03i)T + (33.5 - 58.0i)T^{2}
71 1+(−9.25−2.11i)T+(63.9+30.8i)T2 1 + (-9.25 - 2.11i)T + (63.9 + 30.8i)T^{2}
73 1+(2.30−15.3i)T+(−69.7−21.5i)T2 1 + (2.30 - 15.3i)T + (-69.7 - 21.5i)T^{2}
79 1+(8.66+5.00i)T+(39.5+68.4i)T2 1 + (8.66 + 5.00i)T + (39.5 + 68.4i)T^{2}
83 1+(2.81−3.53i)T+(−18.4−80.9i)T2 1 + (2.81 - 3.53i)T + (-18.4 - 80.9i)T^{2}
89 1+(2.77−1.08i)T+(65.2−60.5i)T2 1 + (2.77 - 1.08i)T + (65.2 - 60.5i)T^{2}
97 1−0.505iT−97T2 1 - 0.505iT - 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

−12.61531666973513792167073376473, −11.49083263486802013568825843948, −10.56536172163568744169804851446, −9.502129699672768707789404043699, −8.565433014983015847210336391562, −8.251012712643700087015518043670, −7.22690788204821036857675180447, −5.16823456062782357440200005853, −3.70750714701103823229869750166, −2.57270604777212526574243901118, 1.13445068121144291336095562312, 3.02339769872739815050535782468, 3.82445548822777585756374273292, 6.88029657804592882718544761341, 7.59310774088054569961109292534, 7.911138547107778871034182968104, 8.960547150788764000199121497980, 10.04029695948633799637872329337, 11.35670717936622720270022608472, 12.16112850918461978176616659966

Graph of the ZZ-function along the critical line