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Properties

Label 2-4608-16.13-c1-0-33
Degree $2$
Conductor $4608$
Sign $0.923 - 0.382i$
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  + (1 − i)5-s + (−2.82 + 2.82i)11-s + (1 + i)13-s + (−2.82 − 2.82i)19-s − 5.65i·23-s + 3i·25-s + (5 + 5i)29-s + 5.65·31-s + (−3 + 3i)37-s + (8.48 − 8.48i)43-s − 11.3·47-s + 7·49-s + (−1 + i)53-s + 5.65i·55-s + (−2.82 + 2.82i)59-s + ⋯
L(s)  = 1  + (0.447 − 0.447i)5-s + (−0.852 + 0.852i)11-s + (0.277 + 0.277i)13-s + (−0.648 − 0.648i)19-s − 1.17i·23-s + 0.600i·25-s + (0.928 + 0.928i)29-s + 1.01·31-s + (−0.493 + 0.493i)37-s + (1.29 − 1.29i)43-s − 1.65·47-s + 49-s + (−0.137 + 0.137i)53-s + 0.762i·55-s + (−0.368 + 0.368i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.875455052\)
\(L(\frac12)\) \(\approx\) \(1.875455052\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + i)T - 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + (2.82 - 2.82i)T - 11iT^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (2.82 + 2.82i)T + 19iT^{2} \)
23 \( 1 + 5.65iT - 23T^{2} \)
29 \( 1 + (-5 - 5i)T + 29iT^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-8.48 + 8.48i)T - 43iT^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 + (2.82 - 2.82i)T - 59iT^{2} \)
61 \( 1 + (-9 - 9i)T + 61iT^{2} \)
67 \( 1 + (-2.82 - 2.82i)T + 67iT^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−8.529056443622386270675677481536, −7.64337701648003325331593061465, −6.83760067303832534062512843180, −6.29649907774467106377837930040, −5.19159610872801571575842512980, −4.83999204402592588878954432488, −3.94965327478409626628828992650, −2.74667059172000300749431376176, −2.07986076010648983654626373238, −0.887930083724822941113372857753, 0.64787524485266178135861110498, 1.99129099954900125542431283423, 2.83429905185183697039904791637, 3.58195462855744339576008492728, 4.58585244713712418371592322585, 5.46796252481233202535944757727, 6.15144580278216787404735587007, 6.59456738137076449781278242368, 7.86462590398509868195650114451, 8.034911908986428362140408270311

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