Properties

Label 2-1148-1148.975-c0-0-1
Degree $2$
Conductor $1148$
Sign $0.489 + 0.872i$
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 + (0.258 − 0.965i)3-s + (0.499 + 0.866i)4-s + (0.866 + 0.5i)5-s + (−0.707 + 0.707i)6-s + (0.965 − 0.258i)7-s − 0.999i·8-s + (−0.499 − 0.866i)10-s + (0.258 − 0.965i)11-s + (0.965 − 0.258i)12-s + (−1 + i)13-s + (−0.965 − 0.258i)14-s + (0.707 − 0.707i)15-s + (−0.5 + 0.866i)16-s + (0.258 + 0.965i)19-s + 0.999i·20-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.258 − 0.965i)3-s + (0.499 + 0.866i)4-s + (0.866 + 0.5i)5-s + (−0.707 + 0.707i)6-s + (0.965 − 0.258i)7-s − 0.999i·8-s + (−0.499 − 0.866i)10-s + (0.258 − 0.965i)11-s + (0.965 − 0.258i)12-s + (−1 + i)13-s + (−0.965 − 0.258i)14-s + (0.707 − 0.707i)15-s + (−0.5 + 0.866i)16-s + (0.258 + 0.965i)19-s + 0.999i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.489 + 0.872i$
Analytic conductor: \(0.572926\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (975, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :0),\ 0.489 + 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9886706586\)
\(L(\frac12)\) \(\approx\) \(0.9886706586\)
\(L(1)\) 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 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 - iT \)
good3 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (1.93 + 0.517i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
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

−9.879764925717354195848787491568, −9.044497099499410923535112720432, −8.173181363787174778634132559584, −7.52187652823058822667690834429, −6.82040042146828233482241038101, −5.99289248705425802870783401262, −4.56314263365796393320914923587, −3.20439896505100334574152553252, −2.02623944933430011140036299855, −1.52157056301630156910381623847, 1.55223270029416349664230584500, 2.62201804816178567106734401254, 4.41339983355262621623509248518, 5.13044657752870464904093981815, 5.71417534006183287830773445977, 7.13089663010296489648446037745, 7.67800044033619870439161042189, 8.903035642866178833048745843870, 9.210211866365503793937327481858, 9.977643053049122521178090112486

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