Properties

Label 2-1148-287.139-c0-0-1
Degree $2$
Conductor $1148$
Sign $0.505 + 0.862i$
Analytic cond. $0.572926$
Root an. cond. $0.756919$
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  i·3-s + (0.587 − 0.809i)5-s + (0.309 + 0.951i)7-s + (0.5 − 0.363i)11-s + (0.951 + 0.309i)13-s + (−0.809 − 0.587i)15-s + (−0.363 − 0.5i)17-s + (−0.951 + 0.309i)19-s + (0.951 − 0.309i)21-s + (−0.5 + 1.53i)23-s i·27-s + (−1.30 − 0.951i)29-s + (0.951 + 1.30i)31-s + (−0.363 − 0.5i)33-s + (0.951 + 0.309i)35-s + ⋯
L(s)  = 1  i·3-s + (0.587 − 0.809i)5-s + (0.309 + 0.951i)7-s + (0.5 − 0.363i)11-s + (0.951 + 0.309i)13-s + (−0.809 − 0.587i)15-s + (−0.363 − 0.5i)17-s + (−0.951 + 0.309i)19-s + (0.951 − 0.309i)21-s + (−0.5 + 1.53i)23-s i·27-s + (−1.30 − 0.951i)29-s + (0.951 + 1.30i)31-s + (−0.363 − 0.5i)33-s + (0.951 + 0.309i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.505 + 0.862i$
Analytic conductor: \(0.572926\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (713, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :0),\ 0.505 + 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.252925742\)
\(L(\frac12)\) \(\approx\) \(1.252925742\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.951 + 0.309i)T \)
good3 \( 1 + iT - T^{2} \)
5 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
11 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
47 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + 0.618iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - iT - T^{2} \)
89 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
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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

−9.530743411663613486758947533379, −9.000522980282069315819318561953, −8.307642249710616038937211568633, −7.43693146472409404172443170018, −6.32994796539499533219158055566, −5.86519209612092490466493687100, −4.86823919753556064750794874024, −3.64648862564269952157790599424, −2.04776698668677107267462544796, −1.45851542342228874076480281989, 1.73059664554291718060909411395, 3.14445561347690398015634708095, 4.19860332853286735152006471386, 4.61656813550390256942635239426, 6.16659891511663796719533445890, 6.56387539676480124568784446204, 7.68590796738333211924994261685, 8.630223042284972389518924295001, 9.554415381699862557615579956319, 10.23228748423377008775007388546

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