Properties

Label 2-1148-1148.319-c0-0-0
Degree $2$
Conductor $1148$
Sign $-0.375 - 0.926i$
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  + (0.866 − 0.5i)2-s + (−0.965 + 0.258i)3-s + (0.499 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.707 + 0.707i)6-s + (−0.258 + 0.965i)7-s − 0.999i·8-s + (−0.499 + 0.866i)10-s + (−0.965 + 0.258i)11-s + (−0.258 + 0.965i)12-s + (−1 + i)13-s + (0.258 + 0.965i)14-s + (0.707 − 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.965 − 0.258i)19-s + 0.999i·20-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.965 + 0.258i)3-s + (0.499 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.707 + 0.707i)6-s + (−0.258 + 0.965i)7-s − 0.999i·8-s + (−0.499 + 0.866i)10-s + (−0.965 + 0.258i)11-s + (−0.258 + 0.965i)12-s + (−1 + i)13-s + (0.258 + 0.965i)14-s + (0.707 − 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.965 − 0.258i)19-s + 0.999i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.375 - 0.926i$
Analytic conductor: \(0.572926\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :0),\ -0.375 - 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4772739620\)
\(L(\frac12)\) \(\approx\) \(0.4772739620\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
41 \( 1 - iT \)
good3 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
5 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.517 - 1.93i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.47573730146293165187468086956, −9.850297727992909066201334399691, −8.713278587403552644932635946345, −7.53024534763845052960659582185, −6.61956438733950205508884901741, −5.92599616863543625768233123805, −4.91142382199333389189604843688, −4.47756490359480918669528487075, −3.06128847188063562535510125837, −2.27197646168893425764201878247, 0.33219174652287564370028545007, 2.69828830385717649051864013198, 3.84193894844243478007508660475, 4.71435147275499014509521915106, 5.42480152044007590392738275229, 6.29944927845401881009477498679, 7.16281922787536654546156758406, 7.893943006390059853696200664485, 8.443752240098103664506686176074, 10.05726141698695993355801994504

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