Properties

Label 2-4608-12.11-c1-0-44
Degree $2$
Conductor $4608$
Sign $0.577 + 0.816i$
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  + 2.82i·5-s − 1.41i·7-s + 2·13-s + 1.41i·17-s − 2.82i·19-s − 6·23-s − 3.00·25-s − 5.65i·29-s − 9.89i·31-s + 4.00·35-s − 2·37-s − 4.24i·41-s − 8.48i·43-s − 10·47-s + 5·49-s + ⋯
L(s)  = 1  + 1.26i·5-s − 0.534i·7-s + 0.554·13-s + 0.342i·17-s − 0.648i·19-s − 1.25·23-s − 0.600·25-s − 1.05i·29-s − 1.77i·31-s + 0.676·35-s − 0.328·37-s − 0.662i·41-s − 1.29i·43-s − 1.45·47-s + 0.714·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (4607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.440204214\)
\(L(\frac12)\) \(\approx\) \(1.440204214\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82iT - 5T^{2} \)
7 \( 1 + 1.41iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 1.41iT - 17T^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + 9.89iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 + 10T + 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 5.65iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 - 12.7iT - 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
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

−8.063185903273047506396163996512, −7.44099731351353017055810124025, −6.75383540289388016043689569726, −6.16362705844462686329400091128, −5.43425506182866440690025483124, −4.11834416624773577878285300205, −3.79972161800046962695423851304, −2.70818312196148287393452785584, −1.96826483377392857615872328186, −0.43000508201211822424306888142, 1.08652884750568271833373000645, 1.86607018937731840724692738260, 3.10480788238594224027819840609, 3.95453226816894290730954599198, 4.88717081131835176986691911032, 5.34930459758865093167425798010, 6.15975675213146004532948389346, 6.93192915184719534918215991765, 7.973585789856338936828036925021, 8.514279705462625924714808563202

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