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Properties

Label 2-2548-364.115-c0-0-1
Degree $2$
Conductor $2548$
Sign $0.469 - 0.882i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (1.17 + 0.315i)5-s + (−0.707 + 0.707i)8-s − 9-s − 1.21·10-s + (0.130 + 0.991i)13-s + (0.500 − 0.866i)16-s + (−0.130 − 0.226i)17-s + (0.965 − 0.258i)18-s + (1.17 − 0.315i)20-s + (0.417 + 0.241i)25-s + (−0.382 − 0.923i)26-s + (0.965 + 1.67i)29-s + (−0.258 + 0.965i)32-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (1.17 + 0.315i)5-s + (−0.707 + 0.707i)8-s − 9-s − 1.21·10-s + (0.130 + 0.991i)13-s + (0.500 − 0.866i)16-s + (−0.130 − 0.226i)17-s + (0.965 − 0.258i)18-s + (1.17 − 0.315i)20-s + (0.417 + 0.241i)25-s + (−0.382 − 0.923i)26-s + (0.965 + 1.67i)29-s + (−0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.469 - 0.882i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ 0.469 - 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8845080800\)
\(L(\frac12)\) \(\approx\) \(0.8845080800\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 \)
13 \( 1 + (-0.130 - 0.991i)T \)
good3 \( 1 + T^{2} \)
5 \( 1 + (-1.17 - 0.315i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 + (0.130 + 0.226i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (-1.91 - 0.513i)T + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + 1.58iT - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + (-1.53 + 0.410i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.739 + 0.198i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.478 + 1.78i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.328725715185375991033112187945, −8.590756428166468550602622797610, −7.81887677470063032085498323470, −6.77009705531337989471085744088, −6.33538322151402968635818369133, −5.63749269547511984806993402419, −4.73322632082958430755032185649, −3.10382365171007611688007048801, −2.38179190039508686334128268930, −1.37327021353875363949167428830, 0.838197630297795961005798730589, 2.20337052246058005025257102211, 2.75741671303235360943586773335, 3.95616203042606142403088058808, 5.38577658142473290528523211958, 5.93047829126975750105486754414, 6.57855021971020209805137877495, 7.79776509473762356698209345536, 8.230388817913373813621441472298, 9.111858363964294754819717477195

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