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Properties

Label 2-468-13.4-c3-0-7
Degree $2$
Conductor $468$
Sign $0.916 + 0.399i$
Analytic cond. $27.6128$
Root an. cond. $5.25479$
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  + 2.49i·5-s + (−15.5 + 8.98i)7-s + (−49.6 − 28.6i)11-s + (46.6 − 4.66i)13-s + (8.75 + 15.1i)17-s + (19.1 − 11.0i)19-s + (65.5 − 113. i)23-s + 118.·25-s + (37.1 − 64.3i)29-s + 110. i·31-s + (−22.4 − 38.8i)35-s + (209. + 120. i)37-s + (369. + 213. i)41-s + (−104. − 180. i)43-s − 158. i·47-s + ⋯
L(s)  = 1  + 0.223i·5-s + (−0.840 + 0.485i)7-s + (−1.35 − 0.784i)11-s + (0.995 − 0.0995i)13-s + (0.124 + 0.216i)17-s + (0.231 − 0.133i)19-s + (0.594 − 1.03i)23-s + 0.950·25-s + (0.237 − 0.411i)29-s + 0.639i·31-s + (−0.108 − 0.187i)35-s + (0.929 + 0.536i)37-s + (1.40 + 0.812i)41-s + (−0.369 − 0.639i)43-s − 0.492i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.916 + 0.399i$
Analytic conductor: \(27.6128\)
Root analytic conductor: \(5.25479\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :3/2),\ 0.916 + 0.399i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.538948186\)
\(L(\frac12)\) \(\approx\) \(1.538948186\)
\(L(\frac{5}{2})\) 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 \( 1 \)
13 \( 1 + (-46.6 + 4.66i)T \)
good5 \( 1 - 2.49iT - 125T^{2} \)
7 \( 1 + (15.5 - 8.98i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (49.6 + 28.6i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-8.75 - 15.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-19.1 + 11.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-65.5 + 113. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-37.1 + 64.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 110. iT - 2.97e4T^{2} \)
37 \( 1 + (-209. - 120. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-369. - 213. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (104. + 180. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 158. iT - 1.03e5T^{2} \)
53 \( 1 + 47.9T + 1.48e5T^{2} \)
59 \( 1 + (-382. + 220. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (434. + 752. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-693. - 400. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-773. + 446. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 413. iT - 3.89e5T^{2} \)
79 \( 1 + 246.T + 4.93e5T^{2} \)
83 \( 1 - 333. iT - 5.71e5T^{2} \)
89 \( 1 + (-328. - 189. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-815. + 470. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−10.68194185783328940310555036779, −9.701918753524552253074244084952, −8.670894911198642442855663152810, −8.001784562537265453212429140033, −6.67212230072217212605913062175, −5.96288443348177479649106172605, −4.91197634719799893267636822338, −3.36592917237725695771296650966, −2.62320242764322851004508953078, −0.65784448068472382438946714947, 0.924981336069861268290621889608, 2.61826571602381849570449944147, 3.75983328860154334116224803633, 4.95945968949462560624737849110, 5.96524516147780394204140313084, 7.10273010136527308888934013439, 7.81732484848157346580593345216, 8.996623607260466054913141209838, 9.810806318879412872268389952939, 10.61649143307394964860450511977

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