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Properties

Label 2-396-11.9-c1-0-3
Degree 22
Conductor 396396
Sign 0.263+0.964i0.263 + 0.964i
Analytic cond. 3.162073.16207
Root an. cond. 1.778221.77822
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  + (−0.190 − 0.587i)5-s + (−2.30 − 1.67i)7-s + (3.23 − 0.726i)11-s + (1.42 − 4.39i)13-s + (−1.42 − 4.39i)17-s + (2.30 − 1.67i)19-s − 6.47·23-s + (3.73 − 2.71i)25-s + (5.16 + 3.75i)29-s + (−1.80 + 5.56i)31-s + (−0.545 + 1.67i)35-s + (−3.92 − 2.85i)37-s + (5.16 − 3.75i)41-s + (2.92 − 2.12i)47-s + (0.354 + 1.08i)49-s + ⋯
L(s)  = 1  + (−0.0854 − 0.262i)5-s + (−0.872 − 0.634i)7-s + (0.975 − 0.219i)11-s + (0.395 − 1.21i)13-s + (−0.346 − 1.06i)17-s + (0.529 − 0.384i)19-s − 1.34·23-s + (0.747 − 0.542i)25-s + (0.958 + 0.696i)29-s + (−0.324 + 0.999i)31-s + (−0.0921 + 0.283i)35-s + (−0.645 − 0.469i)37-s + (0.806 − 0.585i)41-s + (0.426 − 0.310i)47-s + (0.0505 + 0.155i)49-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=((0.263+0.964i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s+1/2) L(s)=((0.263+0.964i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: 0.263+0.964i0.263 + 0.964i
Analytic conductor: 3.162073.16207
Root analytic conductor: 1.778221.77822
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ396(361,⋅)\chi_{396} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 396, ( :1/2), 0.263+0.964i)(2,\ 396,\ (\ :1/2),\ 0.263 + 0.964i)

Particular Values

L(1)L(1) ≈\approx 0.936745−0.715215i0.936745 - 0.715215i
L(12)L(\frac12) ≈\approx 0.936745−0.715215i0.936745 - 0.715215i
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
3 1 1
11 1+(−3.23+0.726i)T 1 + (-3.23 + 0.726i)T
good5 1+(0.190+0.587i)T+(−4.04+2.93i)T2 1 + (0.190 + 0.587i)T + (-4.04 + 2.93i)T^{2}
7 1+(2.30+1.67i)T+(2.16+6.65i)T2 1 + (2.30 + 1.67i)T + (2.16 + 6.65i)T^{2}
13 1+(−1.42+4.39i)T+(−10.5−7.64i)T2 1 + (-1.42 + 4.39i)T + (-10.5 - 7.64i)T^{2}
17 1+(1.42+4.39i)T+(−13.7+9.99i)T2 1 + (1.42 + 4.39i)T + (-13.7 + 9.99i)T^{2}
19 1+(−2.30+1.67i)T+(5.87−18.0i)T2 1 + (-2.30 + 1.67i)T + (5.87 - 18.0i)T^{2}
23 1+6.47T+23T2 1 + 6.47T + 23T^{2}
29 1+(−5.16−3.75i)T+(8.96+27.5i)T2 1 + (-5.16 - 3.75i)T + (8.96 + 27.5i)T^{2}
31 1+(1.80−5.56i)T+(−25.0−18.2i)T2 1 + (1.80 - 5.56i)T + (-25.0 - 18.2i)T^{2}
37 1+(3.92+2.85i)T+(11.4+35.1i)T2 1 + (3.92 + 2.85i)T + (11.4 + 35.1i)T^{2}
41 1+(−5.16+3.75i)T+(12.6−38.9i)T2 1 + (-5.16 + 3.75i)T + (12.6 - 38.9i)T^{2}
43 1+43T2 1 + 43T^{2}
47 1+(−2.92+2.12i)T+(14.5−44.6i)T2 1 + (-2.92 + 2.12i)T + (14.5 - 44.6i)T^{2}
53 1+(2.19−6.74i)T+(−42.8−31.1i)T2 1 + (2.19 - 6.74i)T + (-42.8 - 31.1i)T^{2}
59 1+(8.16+5.93i)T+(18.2+56.1i)T2 1 + (8.16 + 5.93i)T + (18.2 + 56.1i)T^{2}
61 1+(−1.42−4.39i)T+(−49.3+35.8i)T2 1 + (-1.42 - 4.39i)T + (-49.3 + 35.8i)T^{2}
67 1+4.94T+67T2 1 + 4.94T + 67T^{2}
71 1+(−2.66−8.19i)T+(−57.4+41.7i)T2 1 + (-2.66 - 8.19i)T + (-57.4 + 41.7i)T^{2}
73 1+(−9.78−7.10i)T+(22.5+69.4i)T2 1 + (-9.78 - 7.10i)T + (22.5 + 69.4i)T^{2}
79 1+(4.28−13.1i)T+(−63.9−46.4i)T2 1 + (4.28 - 13.1i)T + (-63.9 - 46.4i)T^{2}
83 1+(−4.95−15.2i)T+(−67.1+48.7i)T2 1 + (-4.95 - 15.2i)T + (-67.1 + 48.7i)T^{2}
89 1−8.47T+89T2 1 - 8.47T + 89T^{2}
97 1+(−1.71+5.29i)T+(−78.4−57.0i)T2 1 + (-1.71 + 5.29i)T + (-78.4 - 57.0i)T^{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

−10.99417444892206014286435774181, −10.20903352663760162406501097539, −9.289469587024546057478928894591, −8.431931385448368413251499561505, −7.23201979188717192383296450109, −6.45703604990058265734393677413, −5.27926744391565641597606008954, −3.98733671883468574370914844850, −2.96814174865763253975950217129, −0.816348048498922925798932086761, 1.88735767451518521078901349659, 3.43363927833340893459878816808, 4.40863786205955297066007152770, 6.15560255631355031775978571589, 6.42474598914076281619693840328, 7.77156249221346300027204044322, 8.950218425281242666285468341482, 9.513338313352230665340927047368, 10.53446208597995971564489562773, 11.67459643263741548990241090125

Graph of the ZZ-function along the critical line