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Properties

Label 2-2548-364.107-c0-0-2
Degree $2$
Conductor $2548$
Sign $-0.786 + 0.617i$
Analytic cond. $1.27161$
Root an. cond. $1.12766$
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  − 2-s + 4-s + (0.866 − 1.5i)5-s − 8-s + (−0.5 − 0.866i)9-s + (−0.866 + 1.5i)10-s + (−0.866 + 0.5i)13-s + 16-s − 1.73·17-s + (0.5 + 0.866i)18-s + (0.866 − 1.5i)20-s + (−1 − 1.73i)25-s + (0.866 − 0.5i)26-s + (−0.5 − 0.866i)29-s − 32-s + ⋯
L(s)  = 1  − 2-s + 4-s + (0.866 − 1.5i)5-s − 8-s + (−0.5 − 0.866i)9-s + (−0.866 + 1.5i)10-s + (−0.866 + 0.5i)13-s + 16-s − 1.73·17-s + (0.5 + 0.866i)18-s + (0.866 − 1.5i)20-s + (−1 − 1.73i)25-s + (0.866 − 0.5i)26-s + (−0.5 − 0.866i)29-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.786 + 0.617i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ -0.786 + 0.617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5740216511\)
\(L(\frac12)\) \(\approx\) \(0.5740216511\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 \)
13 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + 1.73T + T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−8.842212610717976035964303595226, −8.426542509389919584659446771266, −7.34508985735582087831145413294, −6.47456834509152382693834946868, −5.86798943675178092538727748228, −4.96519507861112097158669033428, −4.02348382560619562503119063232, −2.51109625564073913110366353361, −1.80436340026683558568075937148, −0.46827535177580317681211945734, 1.93979059056108706555770555828, 2.51833939770882081632799925166, 3.22530757190488570092257881685, 4.84899687526288757266170013470, 5.83664970470651119253871419878, 6.50186286375431648142591456223, 7.15350167324755847219891575323, 7.79330626416206463102464850294, 8.692471934489279560642718212210, 9.472043050950446845281156260617

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