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Properties

Label 2-1148-41.10-c1-0-7
Degree $2$
Conductor $1148$
Sign $0.971 - 0.235i$
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  − 1.21·3-s + (0.00173 + 0.00535i)5-s + (−0.809 + 0.587i)7-s − 1.52·9-s + (0.467 − 1.43i)11-s + (0.600 + 0.436i)13-s + (−0.00211 − 0.00650i)15-s + (1.03 − 3.18i)17-s + (−3.34 + 2.42i)19-s + (0.983 − 0.714i)21-s + (6.46 + 4.70i)23-s + (4.04 − 2.93i)25-s + 5.49·27-s + (0.766 + 2.35i)29-s + (−0.113 + 0.348i)31-s + ⋯
L(s)  = 1  − 0.702·3-s + (0.000777 + 0.00239i)5-s + (−0.305 + 0.222i)7-s − 0.506·9-s + (0.140 − 0.433i)11-s + (0.166 + 0.120i)13-s + (−0.000545 − 0.00168i)15-s + (0.251 − 0.772i)17-s + (−0.766 + 0.557i)19-s + (0.214 − 0.155i)21-s + (1.34 + 0.980i)23-s + (0.809 − 0.587i)25-s + 1.05·27-s + (0.142 + 0.437i)29-s + (−0.0203 + 0.0625i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.971 - 0.235i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.971 - 0.235i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.093767575\)
\(L(\frac12)\) \(\approx\) \(1.093767575\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.688 - 6.36i)T \)
good3 \( 1 + 1.21T + 3T^{2} \)
5 \( 1 + (-0.00173 - 0.00535i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-0.467 + 1.43i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.600 - 0.436i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.03 + 3.18i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.34 - 2.42i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-6.46 - 4.70i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.766 - 2.35i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.113 - 0.348i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.49 + 7.67i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-6.01 - 4.37i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (0.423 + 0.307i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.11 - 3.42i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.58 - 6.23i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.12 + 5.17i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.29 - 7.06i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.00 + 9.25i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 - 15.7T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 + 2.30T + 83T^{2} \)
89 \( 1 + (12.6 - 9.19i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.48 + 7.65i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.823218858582495846602372926393, −9.004424157610934733957130510007, −8.326177043141082276562292791388, −7.17300174512323476092344093645, −6.40360097275620238872209056939, −5.60415953928726990276489933125, −4.88610494699146984795184358155, −3.58892097399441329369449501183, −2.59849721284920753388825107481, −0.876610461837170989158028488032, 0.76065060476920405382709180842, 2.43016084381760913758184916526, 3.60512508820821400118229493441, 4.73986891716192466085848054957, 5.49049488541388800313869969942, 6.55113127202857217137310774862, 6.95165905511391367570465230087, 8.298184000804049812776412319435, 8.856701971310772914544003187182, 9.901844051202713873123238020473

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