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Properties

Label 2-468-117.4-c1-0-5
Degree $2$
Conductor $468$
Sign $0.412 - 0.910i$
Analytic cond. $3.73699$
Root an. cond. $1.93313$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.33 + 1.10i)3-s + (1.5 + 0.866i)5-s + (−0.857 − 0.495i)7-s + (0.571 + 2.94i)9-s + 3.46i·11-s + (−1 + 3.46i)13-s + (1.05 + 2.81i)15-s + (−3.40 − 5.90i)17-s + (6.15 − 3.55i)19-s + (−0.600 − 1.60i)21-s + (2.95 + 5.12i)23-s + (−1 − 1.73i)25-s + (−2.48 + 4.56i)27-s + 8.01·29-s + (−4.86 − 2.81i)31-s + ⋯
L(s)  = 1  + (0.771 + 0.636i)3-s + (0.670 + 0.387i)5-s + (−0.324 − 0.187i)7-s + (0.190 + 0.981i)9-s + 1.04i·11-s + (−0.277 + 0.960i)13-s + (0.271 + 0.725i)15-s + (−0.826 − 1.43i)17-s + (1.41 − 0.814i)19-s + (−0.131 − 0.350i)21-s + (0.616 + 1.06i)23-s + (−0.200 − 0.346i)25-s + (−0.477 + 0.878i)27-s + 1.48·29-s + (−0.874 − 0.504i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.412 - 0.910i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 0.412 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60864 + 1.03718i\)
\(L(\frac12)\) \(\approx\) \(1.60864 + 1.03718i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 - 1.10i)T \)
13 \( 1 + (1 - 3.46i)T \)
good5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.857 + 0.495i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
17 \( 1 + (3.40 + 5.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.15 + 3.55i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.95 - 5.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.01T + 29T^{2} \)
31 \( 1 + (4.86 + 2.81i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.50 + 3.18i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.86 - 10.1i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.15 + 3.55i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + 3.46iT - 59T^{2} \)
61 \( 1 + (-3.50 + 6.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.86 - 1.07i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.15 - 1.81i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.57iT - 73T^{2} \)
79 \( 1 + (-1.85 - 3.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.426 - 0.245i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (13.8 + 7.96i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.9 + 9.20i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.10605835891978229096371572429, −9.827176393998735198002949956482, −9.665316328807317375592566171077, −8.814567118512011723364176404588, −7.29451481995246989743948672098, −6.92681522856887942540803398464, −5.25849565696529553462200237181, −4.45361959620988486993427284330, −3.07849915717207857184986383575, −2.09855915137890387659338451457, 1.22198402725945451509446038428, 2.68039698841170015831730806226, 3.68353687369414867100270016538, 5.39779398607890861303621099042, 6.18376386049813253967667186964, 7.23236682726011238734323050105, 8.464814830257551597545931030765, 8.741503173449202424584982101766, 9.924434762867364413138451675485, 10.67551436460676919487133448025

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