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Properties

Label 2-1148-41.31-c1-0-13
Degree $2$
Conductor $1148$
Sign $0.933 + 0.358i$
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  + 1.49i·3-s + (−0.331 − 1.01i)5-s + (−0.587 − 0.809i)7-s + 0.766·9-s + (2.51 + 0.817i)11-s + (2.13 − 2.93i)13-s + (1.52 − 0.494i)15-s + (−4.41 − 1.43i)17-s + (−4.70 − 6.48i)19-s + (1.20 − 0.878i)21-s + (1.02 + 0.744i)23-s + (3.11 − 2.26i)25-s + 5.62i·27-s + (5.56 − 1.80i)29-s + (−0.118 + 0.364i)31-s + ⋯
L(s)  = 1  + 0.862i·3-s + (−0.148 − 0.455i)5-s + (−0.222 − 0.305i)7-s + 0.255·9-s + (0.759 + 0.246i)11-s + (0.590 − 0.813i)13-s + (0.393 − 0.127i)15-s + (−1.07 − 0.347i)17-s + (−1.08 − 1.48i)19-s + (0.263 − 0.191i)21-s + (0.213 + 0.155i)23-s + (0.623 − 0.452i)25-s + 1.08i·27-s + (1.03 − 0.335i)29-s + (−0.0212 + 0.0653i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.933 + 0.358i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.933 + 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.586197108\)
\(L(\frac12)\) \(\approx\) \(1.586197108\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.68 + 5.81i)T \)
good3 \( 1 - 1.49iT - 3T^{2} \)
5 \( 1 + (0.331 + 1.01i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-2.51 - 0.817i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.13 + 2.93i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.41 + 1.43i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.70 + 6.48i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.02 - 0.744i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-5.56 + 1.80i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.118 - 0.364i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.591 - 1.81i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (1.78 + 1.29i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-4.11 + 5.66i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.335 - 0.109i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-11.8 - 8.58i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-10.6 + 7.73i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.13 - 1.34i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-5.05 - 1.64i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + 3.35T + 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + 1.94T + 83T^{2} \)
89 \( 1 + (-1.27 - 1.75i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.24 + 1.37i)T + (78.4 - 57.0i)T^{2} \)
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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

−9.764610656989828239262250169504, −8.881144722587020737230979203186, −8.503155989135992191757437705074, −7.06890333501730882433898046250, −6.56151843335520054099695694683, −5.22219030719956045978804920527, −4.44096629450112273084476403107, −3.85090845572633425404500743071, −2.52628125577929185276127630789, −0.78778373636600268899263860363, 1.32511506266935226579612799951, 2.32299262471216753378572858900, 3.68635649405712967461836058990, 4.48985560877890590422089831441, 6.09816439925371663943504221569, 6.49331499996743522471419963597, 7.16702942757596937723304979798, 8.292914163532282696748911779694, 8.817272877049546137028455185406, 9.850956130408108802542741581861

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