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Properties

Label 2-1148-41.10-c1-0-17
Degree $2$
Conductor $1148$
Sign $0.311 + 0.950i$
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  + 1.73·3-s + (0.0415 + 0.127i)5-s + (0.809 − 0.587i)7-s + 0.00632·9-s + (1.73 − 5.35i)11-s + (−4.52 − 3.28i)13-s + (0.0720 + 0.221i)15-s + (2.07 − 6.39i)17-s + (−2.86 + 2.07i)19-s + (1.40 − 1.01i)21-s + (−3.50 − 2.54i)23-s + (4.03 − 2.92i)25-s − 5.19·27-s + (2.70 + 8.31i)29-s + (1.43 − 4.41i)31-s + ⋯
L(s)  = 1  + 1.00·3-s + (0.0185 + 0.0571i)5-s + (0.305 − 0.222i)7-s + 0.00210·9-s + (0.524 − 1.61i)11-s + (−1.25 − 0.911i)13-s + (0.0185 + 0.0572i)15-s + (0.504 − 1.55i)17-s + (−0.656 + 0.476i)19-s + (0.306 − 0.222i)21-s + (−0.731 − 0.531i)23-s + (0.806 − 0.585i)25-s − 0.998·27-s + (0.501 + 1.54i)29-s + (0.257 − 0.792i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.311 + 0.950i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.311 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.057684271\)
\(L(\frac12)\) \(\approx\) \(2.057684271\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-6.40 - 0.185i)T \)
good3 \( 1 - 1.73T + 3T^{2} \)
5 \( 1 + (-0.0415 - 0.127i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-1.73 + 5.35i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (4.52 + 3.28i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.07 + 6.39i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.86 - 2.07i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (3.50 + 2.54i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.70 - 8.31i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.43 + 4.41i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.89 - 5.81i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-0.890 - 0.647i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-10.4 - 7.58i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.47 - 4.55i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.05 + 2.21i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.682 - 0.495i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.08 - 3.35i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.02 + 6.22i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 - 1.70T + 73T^{2} \)
79 \( 1 + 6.29T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + (13.2 - 9.61i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.90 + 8.94i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.478082045130913283517424832353, −8.752335084083847357996260339098, −8.065156560922637192879080325315, −7.44368915001283533196067195610, −6.28113256153558866915702972460, −5.36480532966743758508955492604, −4.29019814678104081131573706996, −3.06108959321107291664137082663, −2.63542211012115035207537712208, −0.78319229115878931980127363657, 1.87572535501268341211202117190, 2.43140951271927400081318387021, 3.92596865911587871484312407105, 4.51909323818243068489563862876, 5.73073982588683933835925546815, 6.87064949309857152697873303418, 7.56748857616306182003545285826, 8.374917178109714952888492069721, 9.191078088579525690053315197572, 9.717856004351836187945531065117

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