Properties

Label 2-1148-41.40-c1-0-8
Degree $2$
Conductor $1148$
Sign $0.993 + 0.110i$
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  − 0.162i·3-s − 1.05·5-s + i·7-s + 2.97·9-s − 6.18i·11-s + 5.55i·13-s + 0.171i·15-s + 3.90i·17-s − 2.97i·19-s + 0.162·21-s + 7.17·23-s − 3.87·25-s − 0.968i·27-s − 2.56i·29-s + 0.343·31-s + ⋯
L(s)  = 1  − 0.0935i·3-s − 0.473·5-s + 0.377i·7-s + 0.991·9-s − 1.86i·11-s + 1.53i·13-s + 0.0443i·15-s + 0.947i·17-s − 0.681i·19-s + 0.0353·21-s + 1.49·23-s − 0.775·25-s − 0.186i·27-s − 0.475i·29-s + 0.0616·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.993 + 0.110i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (1065, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.993 + 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.624865208\)
\(L(\frac12)\) \(\approx\) \(1.624865208\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
41 \( 1 + (-0.710 + 6.36i)T \)
good3 \( 1 + 0.162iT - 3T^{2} \)
5 \( 1 + 1.05T + 5T^{2} \)
11 \( 1 + 6.18iT - 11T^{2} \)
13 \( 1 - 5.55iT - 13T^{2} \)
17 \( 1 - 3.90iT - 17T^{2} \)
19 \( 1 + 2.97iT - 19T^{2} \)
23 \( 1 - 7.17T + 23T^{2} \)
29 \( 1 + 2.56iT - 29T^{2} \)
31 \( 1 - 0.343T + 31T^{2} \)
37 \( 1 - 8.19T + 37T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 - 7.52iT - 47T^{2} \)
53 \( 1 + 1.52iT - 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 4.12iT - 67T^{2} \)
71 \( 1 + 7.66iT - 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 + 1.32iT - 89T^{2} \)
97 \( 1 - 1.93iT - 97T^{2} \)
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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

−9.603237800201735451605357451712, −8.952718934386217209663423879687, −8.232706658601049991337170796711, −7.27600506498796629626773923310, −6.46546833820368254246464535957, −5.66010552598103782079570154993, −4.39452327896842654765137465466, −3.73423877647959522413399755975, −2.44349618805520341059300439117, −0.994936003286524678096053701278, 1.03413514005359502483381782368, 2.51092331749428022595212674579, 3.76095853140355128236519614765, 4.60098437537828334117947342760, 5.34745257442012645069869754312, 6.74096889734917518620104559050, 7.48185949996823759944280012539, 7.81940595033776995300879367920, 9.182853955023268668055313816607, 9.962399175988468115474398985851

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