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Properties

Label 2-1148-41.10-c1-0-18
Degree $2$
Conductor $1148$
Sign $-0.538 + 0.842i$
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  + 1.85·3-s + (−1.27 − 3.92i)5-s + (−0.809 + 0.587i)7-s + 0.451·9-s + (1.01 − 3.12i)11-s + (1.36 + 0.993i)13-s + (−2.37 − 7.29i)15-s + (1.46 − 4.51i)17-s + (−4.77 + 3.46i)19-s + (−1.50 + 1.09i)21-s + (3.94 + 2.86i)23-s + (−9.75 + 7.09i)25-s − 4.73·27-s + (−1.01 − 3.13i)29-s + (1.76 − 5.42i)31-s + ⋯
L(s)  = 1  + 1.07·3-s + (−0.570 − 1.75i)5-s + (−0.305 + 0.222i)7-s + 0.150·9-s + (0.305 − 0.941i)11-s + (0.379 + 0.275i)13-s + (−0.612 − 1.88i)15-s + (0.356 − 1.09i)17-s + (−1.09 + 0.795i)19-s + (−0.327 + 0.238i)21-s + (0.822 + 0.597i)23-s + (−1.95 + 1.41i)25-s − 0.911·27-s + (−0.189 − 0.581i)29-s + (0.316 − 0.975i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.538 + 0.842i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.538 + 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.638918065\)
\(L(\frac12)\) \(\approx\) \(1.638918065\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.90 + 5.07i)T \)
good3 \( 1 - 1.85T + 3T^{2} \)
5 \( 1 + (1.27 + 3.92i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-1.01 + 3.12i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.36 - 0.993i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.46 + 4.51i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.77 - 3.46i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-3.94 - 2.86i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.01 + 3.13i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.76 + 5.42i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.62 + 4.99i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (4.26 + 3.10i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-0.898 - 0.652i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.77 + 8.54i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.58 - 4.78i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.303 + 0.220i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.46 - 7.59i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.70 - 11.3i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 2.91T + 73T^{2} \)
79 \( 1 - 6.54T + 79T^{2} \)
83 \( 1 - 7.52T + 83T^{2} \)
89 \( 1 + (-12.9 + 9.40i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.870 - 2.67i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.114867420801706743338192297512, −8.752660654290143493138726793320, −8.207014704507430227289954103052, −7.36984695869329107861570890099, −5.97409388482042298506263016138, −5.19762028383661550002429388462, −4.04032191404813906688904487210, −3.43257192209735042092211491526, −2.02836315135452487844322136308, −0.60631309318528062174177671192, 2.02666145205623098696942116729, 3.07929654636586668879756409870, 3.52005017440716018650072860226, 4.62905729876522000888764173811, 6.36905753723703373105868224413, 6.74641691445723415465331818416, 7.65762234992007579522153955888, 8.342959029894061222865647269312, 9.192772993711842058832799275581, 10.27181120986159670937606025264

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