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Properties

Label 2-1148-41.37-c1-0-10
Degree $2$
Conductor $1148$
Sign $0.712 - 0.701i$
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  + 0.631·3-s + (−0.223 + 0.687i)5-s + (0.809 + 0.587i)7-s − 2.60·9-s + (1.09 + 3.37i)11-s + (4.32 − 3.14i)13-s + (−0.141 + 0.434i)15-s + (−1.57 − 4.83i)17-s + (4.75 + 3.45i)19-s + (0.510 + 0.371i)21-s + (−0.697 + 0.507i)23-s + (3.62 + 2.63i)25-s − 3.53·27-s + (−0.217 + 0.668i)29-s + (2.43 + 7.49i)31-s + ⋯
L(s)  = 1  + 0.364·3-s + (−0.0998 + 0.307i)5-s + (0.305 + 0.222i)7-s − 0.867·9-s + (0.330 + 1.01i)11-s + (1.19 − 0.871i)13-s + (−0.0364 + 0.112i)15-s + (−0.381 − 1.17i)17-s + (1.09 + 0.792i)19-s + (0.111 + 0.0809i)21-s + (−0.145 + 0.105i)23-s + (0.724 + 0.526i)25-s − 0.680·27-s + (−0.0403 + 0.124i)29-s + (0.437 + 1.34i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.712 - 0.701i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.712 - 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.842869590\)
\(L(\frac12)\) \(\approx\) \(1.842869590\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.77 - 2.77i)T \)
good3 \( 1 - 0.631T + 3T^{2} \)
5 \( 1 + (0.223 - 0.687i)T + (-4.04 - 2.93i)T^{2} \)
11 \( 1 + (-1.09 - 3.37i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-4.32 + 3.14i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.57 + 4.83i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.75 - 3.45i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.697 - 0.507i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.217 - 0.668i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.43 - 7.49i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.55 - 10.9i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-0.581 + 0.422i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-0.495 + 0.359i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.703 + 2.16i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.35 + 4.61i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-7.05 - 5.12i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.947 - 2.91i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (4.08 + 12.5i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 - 7.75T + 79T^{2} \)
83 \( 1 - 0.820T + 83T^{2} \)
89 \( 1 + (6.87 + 4.99i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.147 + 0.454i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.865351962774472560107279832067, −8.985479096571415894239364112010, −8.330707713945327023341889304706, −7.49853449711510184333728932924, −6.64452834985741607620128227801, −5.59867454077456905454300060202, −4.82868719239280066615786773480, −3.46528536847091436422078799276, −2.81233626377408738641192498361, −1.34454740389306012835183669614, 0.887080113891122397836193308977, 2.33801440779911198246944155878, 3.58428205161717893269387448525, 4.26303911365830631653072745886, 5.62859937682744559172092902831, 6.19291892274064535584088432153, 7.29882209204976596378321269770, 8.402455578887247021166549434723, 8.693626140973938053402055755691, 9.427591247380871681410857613357

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