Properties

Label 2-4608-12.11-c1-0-60
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.24i·7-s − 4·11-s − 6·13-s − 4.24i·17-s + 2.82i·19-s + 6·23-s + 5·25-s − 8.48i·29-s + 4.24i·31-s − 6·37-s + 1.41i·41-s + 2.82i·43-s − 6·47-s − 10.9·49-s − 8.48i·53-s + ⋯
L(s)  = 1  − 1.60i·7-s − 1.20·11-s − 1.66·13-s − 1.02i·17-s + 0.648i·19-s + 1.25·23-s + 25-s − 1.57i·29-s + 0.762i·31-s − 0.986·37-s + 0.220i·41-s + 0.431i·43-s − 0.875·47-s − 1.57·49-s − 1.16i·53-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: \(1\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
7 \( 1 + 4.24iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 8.48iT - 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 - 2.82iT - 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 8.48iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 4.24iT - 79T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 - 12.7iT - 89T^{2} \)
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

−7.76714131083724239901865740792, −7.02491795510718615019431936981, −6.83405135447474564037169283022, −5.30809244850357323314924184756, −4.96981545298691664253380339263, −4.16502054511091565788199569360, −3.11753878681541518270339046385, −2.43384670690819110329111151351, −1.01465676791758919550831411091, 0, 1.78827932214867660260249596332, 2.68276431274612271923823533354, 3.08596394900680291147734248050, 4.61752563126245202166051772557, 5.21799090855866861698966030986, 5.59738193717085894569396855219, 6.72830861693029482285171102857, 7.28663558619794145335692789488, 8.184119526784080096930054583337

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