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Properties

Label 2-2448-51.50-c0-0-2
Degree $2$
Conductor $2448$
Sign $0.577 + 0.816i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 1.41i·7-s − 11-s − 13-s + 17-s + 19-s + 23-s − 1.41i·31-s − 1.41i·35-s − 1.41i·37-s + 41-s − 43-s + 1.41i·47-s − 1.00·49-s − 1.41i·53-s + ⋯
L(s)  = 1  + 5-s − 1.41i·7-s − 11-s − 13-s + 17-s + 19-s + 23-s − 1.41i·31-s − 1.41i·35-s − 1.41i·37-s + 41-s − 43-s + 1.41i·47-s − 1.00·49-s − 1.41i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.337876305\)
\(L(\frac12)\) \(\approx\) \(1.337876305\)
\(L(1)\) 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 \( 1 \)
17 \( 1 - T \)
good5 \( 1 - T + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.411235665789165665742062316307, −7.916638521008225445748590472741, −7.56772049893492998970091342130, −6.84418102389255540213900169227, −5.72283844985545569111326160417, −5.21274652103574560327797028027, −4.25745655066536670597549280370, −3.18506183361364499007532921764, −2.25157484280692297512002382403, −0.939053410406185639029433388456, 1.59156029747143447406200192966, 2.64983340469225803456731695853, 3.13103680615111031200272645169, 4.97825624274766703740845303727, 5.26446251428881365846986867713, 5.92928189347892353234292917237, 6.92309105069836742310596638135, 7.75843990644232634191827378766, 8.575258828568326335319606212382, 9.318673562884599759903396227912

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