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Properties

Label 2-4608-24.11-c1-0-32
Degree $2$
Conductor $4608$
Sign $0.816 - 0.577i$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
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  + 2.82·5-s + 1.41i·7-s + 2i·13-s + 1.41i·17-s + 2.82·19-s + 6·23-s + 3.00·25-s + 5.65·29-s − 9.89i·31-s + 4.00i·35-s + 2i·37-s + 4.24i·41-s − 8.48·43-s − 10·47-s + 5·49-s + ⋯
L(s)  = 1  + 1.26·5-s + 0.534i·7-s + 0.554i·13-s + 0.342i·17-s + 0.648·19-s + 1.25·23-s + 0.600·25-s + 1.05·29-s − 1.77i·31-s + 0.676i·35-s + 0.328i·37-s + 0.662i·41-s − 1.29·43-s − 1.45·47-s + 0.714·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (2303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.661150393\)
\(L(\frac12)\) \(\approx\) \(2.661150393\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 1.41iT - 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 5.65T + 29T^{2} \)
31 \( 1 + 9.89iT - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 4.24iT - 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 + 10T + 47T^{2} \)
53 \( 1 - 5.65T + 53T^{2} \)
59 \( 1 + 12iT - 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 12.7iT - 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 9.89iT - 89T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
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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

−8.500017587406057626792191331206, −7.72102631175372755554800593861, −6.62492686740849310730390666309, −6.35269923617648039985100612097, −5.37386578189162678006975069305, −4.96057481412867214042343804655, −3.81739057146245058494164130206, −2.76794760390716487654127671663, −2.09108274125028759425641351592, −1.10001293584377675054322218066, 0.838369058726951553592484173434, 1.75495457466366195331429611955, 2.84663025818247163547277017920, 3.48622815491865014169050392003, 4.84353114784319437315436299077, 5.18869902695822134010980756291, 6.07354927287348495114561786600, 6.82969308886975178924486780603, 7.34028740966495056505085870907, 8.380761947456922760715770054216

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