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Properties

Label 2-396-11.4-c3-0-9
Degree $2$
Conductor $396$
Sign $0.905 + 0.424i$
Analytic cond. $23.3647$
Root an. cond. $4.83371$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.7 − 9.98i)5-s + (1.14 + 3.53i)7-s + (35.9 − 6.03i)11-s + (50.6 + 36.8i)13-s + (−67.4 + 49.0i)17-s + (15.3 − 47.3i)19-s + 42.7·23-s + (50.6 − 155. i)25-s + (−47.8 − 147. i)29-s + (163. + 118. i)31-s + (51.1 + 37.1i)35-s + (29.6 + 91.2i)37-s + (139. − 428. i)41-s − 343.·43-s + (−121. + 373. i)47-s + ⋯
L(s)  = 1  + (1.22 − 0.893i)5-s + (0.0620 + 0.190i)7-s + (0.986 − 0.165i)11-s + (1.08 + 0.785i)13-s + (−0.962 + 0.699i)17-s + (0.185 − 0.571i)19-s + 0.387·23-s + (0.404 − 1.24i)25-s + (−0.306 − 0.943i)29-s + (0.947 + 0.688i)31-s + (0.246 + 0.179i)35-s + (0.131 + 0.405i)37-s + (0.529 − 1.63i)41-s − 1.21·43-s + (−0.376 + 1.15i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
Sign: $0.905 + 0.424i$
Analytic conductor: \(23.3647\)
Root analytic conductor: \(4.83371\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{396} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 396,\ (\ :3/2),\ 0.905 + 0.424i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.655067555\)
\(L(\frac12)\) \(\approx\) \(2.655067555\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (-35.9 + 6.03i)T \)
good5 \( 1 + (-13.7 + 9.98i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (-1.14 - 3.53i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (-50.6 - 36.8i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (67.4 - 49.0i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-15.3 + 47.3i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 - 42.7T + 1.21e4T^{2} \)
29 \( 1 + (47.8 + 147. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (-163. - 118. i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (-29.6 - 91.2i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-139. + 428. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 + 343.T + 7.95e4T^{2} \)
47 \( 1 + (121. - 373. i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (155. + 113. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (-163. - 503. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (-273. + 198. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 - 831.T + 3.00e5T^{2} \)
71 \( 1 + (-546. + 397. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (375. + 1.15e3i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (-867. - 630. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (65.4 - 47.5i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 - 30.9T + 7.04e5T^{2} \)
97 \( 1 + (73.7 + 53.5i)T + (2.82e5 + 8.68e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.83360021716887366277270494519, −9.651574286157546454806131340180, −8.987477115217376092559460972881, −8.432943406430149982162140194770, −6.68178629006389395717312831916, −6.11329511841459377378808701068, −4.97943970963629531033501170766, −3.90848551426126947406928322720, −2.12333709562105516026266869217, −1.12105785736983865040003794802, 1.26572061257292023467256296032, 2.58858253386865127187929959712, 3.77658799111766520498093738315, 5.27305910877092778261403336806, 6.33604076593489567960417299618, 6.84771376418211397237195691894, 8.217314869105359288318075677097, 9.304347156860848764044655064374, 10.00073308609013382348173306448, 10.90532425270791541765516366371

Graph of the $Z$-function along the critical line