Properties

Label 2-1148-287.223-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: $\chi_{1148} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :0),\ -0.505 + 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9573646862\)
\(L(\frac12)\) \(\approx\) \(0.9573646862\)
\(L(1)\) 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

−9.733168141825598783917784484550, −8.718705390508558449767906579069, −7.951468554396047307883152262429, −7.06252539669263118385714325967, −6.94807731915831566472753269104, −5.32398952590986787214099128300, −4.55199671937395314639290914545, −3.63110088906770882665329424562, −1.98748383566477736721312006568, −0.913343631389713829830884171142, 2.12432984687920650746321530622, 3.44922150668446790820367215838, 3.92701632411256822637741222932, 5.29979501293574239769257299898, 5.73170310685823106109399453819, 7.23837266657802459782848244315, 7.64199569956829276580802875132, 8.861825623103934583773289478759, 9.567867206998146406909420976364, 10.11461546199930615980998928838

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