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Properties

Label 2-1148-41.40-c1-0-11
Degree $2$
Conductor $1148$
Sign $0.716 + 0.697i$
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  + 2.57i·3-s − 2.38·5-s + i·7-s − 3.60·9-s − 2.88i·11-s − 4.97i·13-s − 6.13i·15-s − 7.25i·17-s + 4.84i·19-s − 2.57·21-s − 1.50·23-s + 0.688·25-s − 1.56i·27-s − 4.90i·29-s − 5.66·31-s + ⋯
L(s)  = 1  + 1.48i·3-s − 1.06·5-s + 0.377i·7-s − 1.20·9-s − 0.869i·11-s − 1.37i·13-s − 1.58i·15-s − 1.75i·17-s + 1.11i·19-s − 0.561·21-s − 0.312·23-s + 0.137·25-s − 0.301i·27-s − 0.910i·29-s − 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.716 + 0.697i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.716 + 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7246036792\)
\(L(\frac12)\) \(\approx\) \(0.7246036792\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.46 + 4.58i)T \)
good3 \( 1 - 2.57iT - 3T^{2} \)
5 \( 1 + 2.38T + 5T^{2} \)
11 \( 1 + 2.88iT - 11T^{2} \)
13 \( 1 + 4.97iT - 13T^{2} \)
17 \( 1 + 7.25iT - 17T^{2} \)
19 \( 1 - 4.84iT - 19T^{2} \)
23 \( 1 + 1.50T + 23T^{2} \)
29 \( 1 + 4.90iT - 29T^{2} \)
31 \( 1 + 5.66T + 31T^{2} \)
37 \( 1 - 9.73T + 37T^{2} \)
43 \( 1 + 4.00T + 43T^{2} \)
47 \( 1 - 8.40iT - 47T^{2} \)
53 \( 1 + 5.39iT - 53T^{2} \)
59 \( 1 - 9.99T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 1.61iT - 67T^{2} \)
71 \( 1 + 15.4iT - 71T^{2} \)
73 \( 1 + 1.66T + 73T^{2} \)
79 \( 1 + 13.1iT - 79T^{2} \)
83 \( 1 + 7.34T + 83T^{2} \)
89 \( 1 - 6.68iT - 89T^{2} \)
97 \( 1 + 3.42iT - 97T^{2} \)
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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

−9.701277215732169091531193810745, −9.044650346262631876752614820251, −8.044687139536480278454368852942, −7.59635499958655036118088180363, −5.98969623502257424185111639390, −5.32728561995550493900655370623, −4.37953142299310830130381982472, −3.55460343834136696400349884037, −2.85916365224108156051093640185, −0.33571481423076726880539739039, 1.33016825455482123900322005766, 2.29702506255310730002715441722, 3.82909396065556690523374870687, 4.52622617603644545447977740209, 6.00567270410343681817788092452, 6.91596799415291572129864308818, 7.25754954412240949999369545853, 8.083767108397550128751925010574, 8.785861943722633825549134708783, 9.857809774208018435511650224043

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