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Properties

Label 2-14e2-196.59-c1-0-5
Degree 22
Conductor 196196
Sign −0.455−0.890i-0.455 - 0.890i
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  + (−1.02 + 0.979i)2-s + (−1.63 + 1.11i)3-s + (0.0826 − 1.99i)4-s + (2.43 − 0.182i)5-s + (0.577 − 2.74i)6-s + (1.91 + 1.82i)7-s + (1.87 + 2.12i)8-s + (0.337 − 0.859i)9-s + (−2.30 + 2.57i)10-s + (−2.09 + 0.823i)11-s + (2.09 + 3.36i)12-s + (0.425 + 0.339i)13-s + (−3.74 + 0.00463i)14-s + (−3.78 + 3.01i)15-s + (−3.98 − 0.330i)16-s + (−2.31 + 2.49i)17-s + ⋯
L(s)  = 1  + (−0.721 + 0.692i)2-s + (−0.944 + 0.644i)3-s + (0.0413 − 0.999i)4-s + (1.08 − 0.0816i)5-s + (0.235 − 1.11i)6-s + (0.722 + 0.691i)7-s + (0.661 + 0.749i)8-s + (0.112 − 0.286i)9-s + (−0.729 + 0.813i)10-s + (−0.632 + 0.248i)11-s + (0.604 + 0.970i)12-s + (0.117 + 0.0940i)13-s + (−0.999 + 0.00123i)14-s + (−0.976 + 0.779i)15-s + (−0.996 − 0.0825i)16-s + (−0.562 + 0.606i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((−0.455−0.890i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((−0.455−0.890i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −0.455−0.890i-0.455 - 0.890i
Analytic conductor: 1.565061.56506
Root analytic conductor: 1.251021.25102
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ196(59,⋅)\chi_{196} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 196, ( :1/2), −0.455−0.890i)(2,\ 196,\ (\ :1/2),\ -0.455 - 0.890i)

Particular Values

L(1)L(1) ≈\approx 0.377315+0.616698i0.377315 + 0.616698i
L(12)L(\frac12) ≈\approx 0.377315+0.616698i0.377315 + 0.616698i
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+(1.02−0.979i)T 1 + (1.02 - 0.979i)T
7 1+(−1.91−1.82i)T 1 + (-1.91 - 1.82i)T
good3 1+(1.63−1.11i)T+(1.09−2.79i)T2 1 + (1.63 - 1.11i)T + (1.09 - 2.79i)T^{2}
5 1+(−2.43+0.182i)T+(4.94−0.745i)T2 1 + (-2.43 + 0.182i)T + (4.94 - 0.745i)T^{2}
11 1+(2.09−0.823i)T+(8.06−7.48i)T2 1 + (2.09 - 0.823i)T + (8.06 - 7.48i)T^{2}
13 1+(−0.425−0.339i)T+(2.89+12.6i)T2 1 + (-0.425 - 0.339i)T + (2.89 + 12.6i)T^{2}
17 1+(2.31−2.49i)T+(−1.27−16.9i)T2 1 + (2.31 - 2.49i)T + (-1.27 - 16.9i)T^{2}
19 1+(0.309+0.535i)T+(−9.5+16.4i)T2 1 + (0.309 + 0.535i)T + (-9.5 + 16.4i)T^{2}
23 1+(−5.40−5.82i)T+(−1.71+22.9i)T2 1 + (-5.40 - 5.82i)T + (-1.71 + 22.9i)T^{2}
29 1+(1.87−8.19i)T+(−26.1−12.5i)T2 1 + (1.87 - 8.19i)T + (-26.1 - 12.5i)T^{2}
31 1+(−0.779+1.35i)T+(−15.5−26.8i)T2 1 + (-0.779 + 1.35i)T + (-15.5 - 26.8i)T^{2}
37 1+(−11.4+3.53i)T+(30.5−20.8i)T2 1 + (-11.4 + 3.53i)T + (30.5 - 20.8i)T^{2}
41 1+(0.189−0.393i)T+(−25.5−32.0i)T2 1 + (0.189 - 0.393i)T + (-25.5 - 32.0i)T^{2}
43 1+(2.48+5.15i)T+(−26.8+33.6i)T2 1 + (2.48 + 5.15i)T + (-26.8 + 33.6i)T^{2}
47 1+(2.28+0.344i)T+(44.9+13.8i)T2 1 + (2.28 + 0.344i)T + (44.9 + 13.8i)T^{2}
53 1+(−1.32−0.407i)T+(43.7+29.8i)T2 1 + (-1.32 - 0.407i)T + (43.7 + 29.8i)T^{2}
59 1+(−0.851+11.3i)T+(−58.3−8.79i)T2 1 + (-0.851 + 11.3i)T + (-58.3 - 8.79i)T^{2}
61 1+(4.06+13.1i)T+(−50.4+34.3i)T2 1 + (4.06 + 13.1i)T + (-50.4 + 34.3i)T^{2}
67 1+(5.00+2.89i)T+(33.5+58.0i)T2 1 + (5.00 + 2.89i)T + (33.5 + 58.0i)T^{2}
71 1+(−9.15+2.08i)T+(63.9−30.8i)T2 1 + (-9.15 + 2.08i)T + (63.9 - 30.8i)T^{2}
73 1+(−0.0617−0.409i)T+(−69.7+21.5i)T2 1 + (-0.0617 - 0.409i)T + (-69.7 + 21.5i)T^{2}
79 1+(−5.72+3.30i)T+(39.5−68.4i)T2 1 + (-5.72 + 3.30i)T + (39.5 - 68.4i)T^{2}
83 1+(−5.68−7.12i)T+(−18.4+80.9i)T2 1 + (-5.68 - 7.12i)T + (-18.4 + 80.9i)T^{2}
89 1+(−2.24−0.879i)T+(65.2+60.5i)T2 1 + (-2.24 - 0.879i)T + (65.2 + 60.5i)T^{2}
97 1+16.6iT−97T2 1 + 16.6iT - 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

−12.87349876005850017503366856722, −11.27853298618573098736480139493, −10.84151785131566691279195658692, −9.764013910537154760720441441503, −9.023257226873562151266832124272, −7.81681829774802727799471950698, −6.37110850401857206187662674471, −5.45284799855962929190333534901, −4.92097194907541784584189748568, −1.94539781430128658757605064105, 0.955856899306573965428602064407, 2.50963899366327486340994039708, 4.62331023946777626850552918624, 6.04477210281411513909401175764, 7.07404727994871619491724739312, 8.162180771656028572976941793899, 9.418188098331164778265459921714, 10.44504713951519309750733650479, 11.13257333211896226609910654397, 11.91035175257004534180725405793

Graph of the ZZ-function along the critical line