Properties

Label 2-4608-16.5-c1-0-59
Degree $2$
Conductor $4608$
Sign $-0.707 + 0.707i$
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  + (−2.41 − 2.41i)5-s − 1.53i·7-s + (3.37 + 3.37i)11-s + (0.414 − 0.414i)13-s − 2.82·17-s + (0.317 − 0.317i)19-s − 5.86i·23-s + 6.65i·25-s + (3.24 − 3.24i)29-s + 7.39·31-s + (−3.69 + 3.69i)35-s + (−3.58 − 3.58i)37-s − 4i·41-s + (−1.84 − 1.84i)43-s + 7.39·47-s + ⋯
L(s)  = 1  + (−1.07 − 1.07i)5-s − 0.578i·7-s + (1.01 + 1.01i)11-s + (0.114 − 0.114i)13-s − 0.685·17-s + (0.0727 − 0.0727i)19-s − 1.22i·23-s + 1.33i·25-s + (0.602 − 0.602i)29-s + 1.32·31-s + (−0.624 + 0.624i)35-s + (−0.589 − 0.589i)37-s − 0.624i·41-s + (−0.281 − 0.281i)43-s + 1.07·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (3457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.162900645\)
\(L(\frac12)\) \(\approx\) \(1.162900645\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.41 + 2.41i)T + 5iT^{2} \)
7 \( 1 + 1.53iT - 7T^{2} \)
11 \( 1 + (-3.37 - 3.37i)T + 11iT^{2} \)
13 \( 1 + (-0.414 + 0.414i)T - 13iT^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + (-0.317 + 0.317i)T - 19iT^{2} \)
23 \( 1 + 5.86iT - 23T^{2} \)
29 \( 1 + (-3.24 + 3.24i)T - 29iT^{2} \)
31 \( 1 - 7.39T + 31T^{2} \)
37 \( 1 + (3.58 + 3.58i)T + 37iT^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 + (1.84 + 1.84i)T + 43iT^{2} \)
47 \( 1 - 7.39T + 47T^{2} \)
53 \( 1 + (-5.24 - 5.24i)T + 53iT^{2} \)
59 \( 1 + (-1.84 - 1.84i)T + 59iT^{2} \)
61 \( 1 + (9.24 - 9.24i)T - 61iT^{2} \)
67 \( 1 + (7.07 - 7.07i)T - 67iT^{2} \)
71 \( 1 + 11.9iT - 71T^{2} \)
73 \( 1 + 10.4iT - 73T^{2} \)
79 \( 1 + 6.12T + 79T^{2} \)
83 \( 1 + (-2.48 + 2.48i)T - 83iT^{2} \)
89 \( 1 - 0.828iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−8.057089312386634577832817772927, −7.34005422444555565566636421666, −6.78271518965496887067031199803, −5.87877332386198114727329611294, −4.61113150178944075821419606803, −4.44613269311099368338738908283, −3.79947150252392346967911430929, −2.50987456814370423180048380665, −1.28688801243571720454641933233, −0.38280853569707423071652775847, 1.15868430742329960546886703553, 2.53925616891296166537170518390, 3.32565087695571024815320955614, 3.85108961508792794142348350679, 4.79784346011652978234276523661, 5.86044182627609456022079909138, 6.52428669895417413634555793946, 7.03782807752930849214312981330, 7.902801479486522916507383249227, 8.548696307787177407049321673746

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