Properties

Label 2-4608-12.11-c1-0-10
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  + 3.86i·5-s − 2.44i·7-s + 5.46·11-s − 2·13-s − 6.31i·17-s + 4.89i·19-s − 7.46·23-s − 9.92·25-s + 1.03i·29-s + 8.10i·31-s + 9.46·35-s + 0.535·37-s + 9.14i·41-s + 2.82i·43-s − 0.535·47-s + ⋯
L(s)  = 1  + 1.72i·5-s − 0.925i·7-s + 1.64·11-s − 0.554·13-s − 1.53i·17-s + 1.12i·19-s − 1.55·23-s − 1.98·25-s + 0.192i·29-s + 1.45i·31-s + 1.59·35-s + 0.0881·37-s + 1.42i·41-s + 0.431i·43-s − 0.0781·47-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

\(L(1)\) \(\approx\) \(1.425577254\)
\(L(\frac12)\) \(\approx\) \(1.425577254\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.86iT - 5T^{2} \)
7 \( 1 + 2.44iT - 7T^{2} \)
11 \( 1 - 5.46T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 6.31iT - 17T^{2} \)
19 \( 1 - 4.89iT - 19T^{2} \)
23 \( 1 + 7.46T + 23T^{2} \)
29 \( 1 - 1.03iT - 29T^{2} \)
31 \( 1 - 8.10iT - 31T^{2} \)
37 \( 1 - 0.535T + 37T^{2} \)
41 \( 1 - 9.14iT - 41T^{2} \)
43 \( 1 - 2.82iT - 43T^{2} \)
47 \( 1 + 0.535T + 47T^{2} \)
53 \( 1 + 3.10iT - 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 13.3iT - 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 1.69iT - 79T^{2} \)
83 \( 1 + 6.53T + 83T^{2} \)
89 \( 1 - 1.41iT - 89T^{2} \)
97 \( 1 - 6.92T + 97T^{2} \)
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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

−8.451673885577523250166127507924, −7.55094923407555772454010229189, −7.10121757772562017628123623221, −6.55001561329794908816631948705, −5.94438847885873480554388822103, −4.70934245813650584462135966771, −3.84097489174161563077730888957, −3.34118880527192506039179513738, −2.38225502175604994893684056929, −1.26082439128030447426627861743, 0.39986872147877934319741598242, 1.61015289082083848118800461316, 2.24263966252081181739439754305, 3.86600021733841828665965406267, 4.21833940890494343050004462268, 5.12401670393467489300997216780, 5.88789932727723607978869937357, 6.34291927073337813234853426991, 7.50810773401113630285391136446, 8.333529720993723531381322962543

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