Properties

Label 2-1148-1148.655-c0-0-5
Degree $2$
Conductor $1148$
Sign $0.750 - 0.660i$
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  + (0.5 + 0.866i)2-s + (0.965 − 1.67i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 1.5i)5-s + 1.93·6-s + (−0.258 + 0.965i)7-s − 0.999·8-s + (−1.36 − 2.36i)9-s + (−0.866 + 1.5i)10-s + (0.258 − 0.448i)11-s + (0.965 + 1.67i)12-s + (−0.965 + 0.258i)14-s + 3.34·15-s + (−0.5 − 0.866i)16-s + (1.36 − 2.36i)18-s + (−0.258 − 0.448i)19-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.965 − 1.67i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 1.5i)5-s + 1.93·6-s + (−0.258 + 0.965i)7-s − 0.999·8-s + (−1.36 − 2.36i)9-s + (−0.866 + 1.5i)10-s + (0.258 − 0.448i)11-s + (0.965 + 1.67i)12-s + (−0.965 + 0.258i)14-s + 3.34·15-s + (−0.5 − 0.866i)16-s + (1.36 − 2.36i)18-s + (−0.258 − 0.448i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.750 - 0.660i$
Analytic conductor: \(0.572926\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :0),\ 0.750 - 0.660i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.809729244\)
\(L(\frac12)\) \(\approx\) \(1.809729244\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
41 \( 1 - T \)
good3 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.93T + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−9.644563265218443843544651010613, −9.006882190222763517654815996452, −8.232629488128721907264115632162, −7.40899424755949945930532328517, −6.62059304672047393159878811757, −6.31978175981018118736023005523, −5.54254930400929333986586177999, −3.50189059843053781102807040703, −2.81962520485208165693112483111, −2.10764208365349026658219098080, 1.59438215829365326960239542714, 2.82038984406950722036720502042, 3.99918958122409024678437666689, 4.42470825690778690460718283740, 5.13911413094064660381673666426, 6.04116061138301736448240791473, 7.86403190659206193508086470802, 8.854092873896675216444825801501, 9.272396752218240182718267893067, 9.964040532992200261950076304301

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