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Properties

Label 2-1148-41.32-c1-0-6
Degree $2$
Conductor $1148$
Sign $-0.241 - 0.970i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.96 + 1.96i)3-s − 0.264i·5-s + (0.707 − 0.707i)7-s − 4.69i·9-s + (1.34 − 1.34i)11-s + (−2.27 + 2.27i)13-s + (0.519 + 0.519i)15-s + (0.388 + 0.388i)17-s + (1.69 + 1.69i)19-s + 2.77i·21-s + 8.48·23-s + 4.92·25-s + (3.33 + 3.33i)27-s + (−4.81 + 4.81i)29-s + 2.85·31-s + ⋯
L(s)  = 1  + (−1.13 + 1.13i)3-s − 0.118i·5-s + (0.267 − 0.267i)7-s − 1.56i·9-s + (0.406 − 0.406i)11-s + (−0.629 + 0.629i)13-s + (0.134 + 0.134i)15-s + (0.0941 + 0.0941i)17-s + (0.389 + 0.389i)19-s + 0.605i·21-s + 1.76·23-s + 0.985·25-s + (0.641 + 0.641i)27-s + (−0.894 + 0.894i)29-s + 0.512·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.241 - 0.970i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9868420915\)
\(L(\frac12)\) \(\approx\) \(0.9868420915\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (1.29 - 6.27i)T \)
good3 \( 1 + (1.96 - 1.96i)T - 3iT^{2} \)
5 \( 1 + 0.264iT - 5T^{2} \)
11 \( 1 + (-1.34 + 1.34i)T - 11iT^{2} \)
13 \( 1 + (2.27 - 2.27i)T - 13iT^{2} \)
17 \( 1 + (-0.388 - 0.388i)T + 17iT^{2} \)
19 \( 1 + (-1.69 - 1.69i)T + 19iT^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 + (4.81 - 4.81i)T - 29iT^{2} \)
31 \( 1 - 2.85T + 31T^{2} \)
37 \( 1 + 5.41T + 37T^{2} \)
43 \( 1 + 4.60iT - 43T^{2} \)
47 \( 1 + (-4.98 - 4.98i)T + 47iT^{2} \)
53 \( 1 + (1.45 - 1.45i)T - 53iT^{2} \)
59 \( 1 + 7.20T + 59T^{2} \)
61 \( 1 - 7.94iT - 61T^{2} \)
67 \( 1 + (-2.37 - 2.37i)T + 67iT^{2} \)
71 \( 1 + (10.1 - 10.1i)T - 71iT^{2} \)
73 \( 1 - 4.71iT - 73T^{2} \)
79 \( 1 + (-2.09 + 2.09i)T - 79iT^{2} \)
83 \( 1 - 5.22T + 83T^{2} \)
89 \( 1 + (8.76 - 8.76i)T - 89iT^{2} \)
97 \( 1 + (-1.75 - 1.75i)T + 97iT^{2} \)
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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

−10.18595716818332195194460072261, −9.303484051135791019231579421479, −8.745413309451071678988695288120, −7.35250431651174217388466494331, −6.61173606220496994228588663738, −5.54736866928595491369397753487, −4.93321108644575530541452621963, −4.17466887315521591245069903208, −3.11451378307140158113392950986, −1.16164015866505977995870036339, 0.59730238722544375337206945198, 1.82613097455972437271740911379, 3.06674687855330956507930677082, 4.76219091856951205454929026727, 5.34860583195494977838045758601, 6.27842297135186902757765938922, 7.12568214752546434489895391566, 7.51436580748363222855969086034, 8.674274136266823493239263290713, 9.592981744468666619053270565806

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