Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.900 + 0.434i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.68 + 1.56i)2-s + (11.1 − 11.5i)4-s + (−1.01 − 1.75i)5-s + (20.0 + 11.5i)7-s + (−22.8 + 59.7i)8-s + (6.48 + 4.88i)10-s + (−4.32 − 2.49i)11-s + (−137. − 238. i)13-s + (−91.8 − 11.2i)14-s + (−9.35 − 255. i)16-s + 266.·17-s + 367. i·19-s + (−31.5 − 7.83i)20-s + (19.8 + 2.42i)22-s + (544. − 314. i)23-s + ⋯
L(s)  = 1  + (−0.920 + 0.391i)2-s + (0.694 − 0.719i)4-s + (−0.0406 − 0.0703i)5-s + (0.408 + 0.236i)7-s + (−0.357 + 0.934i)8-s + (0.0648 + 0.0488i)10-s + (−0.0357 − 0.0206i)11-s + (−0.815 − 1.41i)13-s + (−0.468 − 0.0573i)14-s + (−0.0365 − 0.999i)16-s + 0.920·17-s + 1.01i·19-s + (−0.0788 − 0.0195i)20-s + (0.0410 + 0.00501i)22-s + (1.02 − 0.594i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.900 + 0.434i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.900 + 0.434i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.07275 - 0.245262i\)
\(L(\frac12)\)  \(\approx\)  \(1.07275 - 0.245262i\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (3.68 - 1.56i)T \)
3 \( 1 \)
good5 \( 1 + (1.01 + 1.75i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (-20.0 - 11.5i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (4.32 + 2.49i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (137. + 238. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 266.T + 8.35e4T^{2} \)
19 \( 1 - 367. iT - 1.30e5T^{2} \)
23 \( 1 + (-544. + 314. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-319. + 553. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-1.19e3 + 687. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 1.46e3T + 1.87e6T^{2} \)
41 \( 1 + (593. + 1.02e3i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-1.43e3 - 825. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (307. + 177. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 5.29e3T + 7.89e6T^{2} \)
59 \( 1 + (-5.22e3 + 3.01e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (833. - 1.44e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (1.90e3 - 1.10e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 524. iT - 2.54e7T^{2} \)
73 \( 1 + 1.49e3T + 2.83e7T^{2} \)
79 \( 1 + (-4.44e3 - 2.56e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (6.91e3 + 3.99e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 8.86e3T + 6.27e7T^{2} \)
97 \( 1 + (3.40e3 - 5.90e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.72547407543930725013672760328, −11.73388284079631756126121446700, −10.47243220206186918480885527463, −9.767439152917441884811861686533, −8.310321654435023849685686813942, −7.71879623852163485325016978480, −6.20475879083290786807593261614, −5.03911184885412605977723575961, −2.67656354170817125303490563006, −0.77603793797390430330979597529, 1.27244621694711728187732417212, 2.94502735140006268417057326160, 4.71952512639696909448590351340, 6.73894259457453846046422527386, 7.59514199875557197568225539452, 8.921985504701647228523681292063, 9.748686801239933376831731514516, 10.99202614232278023631433990046, 11.71740758548666401961311015680, 12.79375814161301929220597496858

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