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Properties

Label 2-1148-1148.655-c0-0-3
Degree $2$
Conductor $1148$
Sign $0.997 + 0.0633i$
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 + (−0.866 + 1.5i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 1.73·6-s + (−0.866 + 0.5i)7-s + 0.999·8-s + (−1 − 1.73i)9-s + (−0.499 + 0.866i)10-s + (0.866 − 1.5i)11-s + (−0.866 − 1.49i)12-s + (0.866 + 0.499i)14-s + 1.73·15-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)18-s + (0.866 + 1.5i)19-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.866 + 1.5i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 1.73·6-s + (−0.866 + 0.5i)7-s + 0.999·8-s + (−1 − 1.73i)9-s + (−0.499 + 0.866i)10-s + (0.866 − 1.5i)11-s + (−0.866 − 1.49i)12-s + (0.866 + 0.499i)14-s + 1.73·15-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)18-s + (0.866 + 1.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.997 + 0.0633i$
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: \((2,\ 1148,\ (\ :0),\ 0.997 + 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4750862991\)
\(L(\frac12)\) \(\approx\) \(0.4750862991\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 - T \)
good3 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.937385756366264268940663503496, −9.383480898375803178903597926858, −8.754498856670936335915056239115, −8.020741592860223250133033168741, −6.32183502768526772817521855746, −5.59119761805358063146580390202, −4.57877237117567901610476015304, −3.73755061338773123451720899785, −3.18223877354386093336492984752, −0.867940720250004932536662816770, 0.890335062248904995346661530888, 2.37231380121655745461670150510, 4.07299203410147548833264677647, 5.26733591644617900756007867152, 6.30667867369728698554807599351, 7.00899317833891557675586239882, 7.11096120418560520981623283045, 7.82369188836759310891985866215, 9.223163142563012697096821927109, 9.839360408104940067195377931498

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