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Properties

Label 2-1148-41.31-c1-0-4
Degree $2$
Conductor $1148$
Sign $-0.623 - 0.782i$
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  + 2.20i·3-s + (0.330 + 1.01i)5-s + (−0.587 − 0.809i)7-s − 1.86·9-s + (5.31 + 1.72i)11-s + (−2.00 + 2.76i)13-s + (−2.23 + 0.727i)15-s + (3.76 + 1.22i)17-s + (1.02 + 1.40i)19-s + (1.78 − 1.29i)21-s + (−3.68 − 2.67i)23-s + (3.12 − 2.26i)25-s + 2.50i·27-s + (−7.36 + 2.39i)29-s + (−0.928 + 2.85i)31-s + ⋯
L(s)  = 1  + 1.27i·3-s + (0.147 + 0.454i)5-s + (−0.222 − 0.305i)7-s − 0.620·9-s + (1.60 + 0.520i)11-s + (−0.557 + 0.766i)13-s + (−0.578 + 0.187i)15-s + (0.912 + 0.296i)17-s + (0.234 + 0.322i)19-s + (0.389 − 0.282i)21-s + (−0.768 − 0.558i)23-s + (0.624 − 0.453i)25-s + 0.482i·27-s + (−1.36 + 0.444i)29-s + (−0.166 + 0.513i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.623 - 0.782i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.623 - 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.663789547\)
\(L(\frac12)\) \(\approx\) \(1.663789547\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (5.26 - 3.64i)T \)
good3 \( 1 - 2.20iT - 3T^{2} \)
5 \( 1 + (-0.330 - 1.01i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-5.31 - 1.72i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (2.00 - 2.76i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.76 - 1.22i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.02 - 1.40i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.68 + 2.67i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (7.36 - 2.39i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.928 - 2.85i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.05 - 3.26i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-4.90 - 3.56i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-6.27 + 8.63i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-11.9 + 3.87i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.16 + 5.20i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.86 - 2.07i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (12.6 - 4.09i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (13.7 + 4.46i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 + 12.4iT - 79T^{2} \)
83 \( 1 + 4.53T + 83T^{2} \)
89 \( 1 + (-9.01 - 12.4i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-15.9 + 5.18i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.28262412713462930114957071868, −9.325464596926504290357816229613, −8.891142363641064139877919916621, −7.48699034351707878599088929333, −6.75957962924384430385824200089, −5.85334876788808164411736638084, −4.66912640903307437521392552960, −4.02436845341073623551446415031, −3.24923172785990687966684583774, −1.67382531884284524991694178608, 0.77659028692303743988315068491, 1.76519932395027886554001056010, 3.04701881069545670855837678382, 4.19338078402947363279552622455, 5.69702104508942012702261104102, 5.97671440826203351766090249848, 7.34925173567537357163591805549, 7.47772700423976180321455319523, 8.782533142308026621499239962408, 9.263589108935329155532641583487

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