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Properties

Label 2-1148-41.32-c1-0-3
Degree $2$
Conductor $1148$
Sign $-0.430 - 0.902i$
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  + (0.147 − 0.147i)3-s − 2.61i·5-s + (0.707 − 0.707i)7-s + 2.95i·9-s + (−3.93 + 3.93i)11-s + (−4.00 + 4.00i)13-s + (−0.384 − 0.384i)15-s + (−3.80 − 3.80i)17-s + (2.63 + 2.63i)19-s − 0.207i·21-s − 6.55·23-s − 1.84·25-s + (0.875 + 0.875i)27-s + (−5.02 + 5.02i)29-s + 5.50·31-s + ⋯
L(s)  = 1  + (0.0848 − 0.0848i)3-s − 1.16i·5-s + (0.267 − 0.267i)7-s + 0.985i·9-s + (−1.18 + 1.18i)11-s + (−1.11 + 1.11i)13-s + (−0.0993 − 0.0993i)15-s + (−0.923 − 0.923i)17-s + (0.604 + 0.604i)19-s − 0.0453i·21-s − 1.36·23-s − 0.368·25-s + (0.168 + 0.168i)27-s + (−0.933 + 0.933i)29-s + 0.988·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.430 - 0.902i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.430 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6299884928\)
\(L(\frac12)\) \(\approx\) \(0.6299884928\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (0.0213 - 6.40i)T \)
good3 \( 1 + (-0.147 + 0.147i)T - 3iT^{2} \)
5 \( 1 + 2.61iT - 5T^{2} \)
11 \( 1 + (3.93 - 3.93i)T - 11iT^{2} \)
13 \( 1 + (4.00 - 4.00i)T - 13iT^{2} \)
17 \( 1 + (3.80 + 3.80i)T + 17iT^{2} \)
19 \( 1 + (-2.63 - 2.63i)T + 19iT^{2} \)
23 \( 1 + 6.55T + 23T^{2} \)
29 \( 1 + (5.02 - 5.02i)T - 29iT^{2} \)
31 \( 1 - 5.50T + 31T^{2} \)
37 \( 1 - 3.96T + 37T^{2} \)
43 \( 1 + 5.74iT - 43T^{2} \)
47 \( 1 + (-3.50 - 3.50i)T + 47iT^{2} \)
53 \( 1 + (5.19 - 5.19i)T - 53iT^{2} \)
59 \( 1 - 6.18T + 59T^{2} \)
61 \( 1 - 1.26iT - 61T^{2} \)
67 \( 1 + (6.45 + 6.45i)T + 67iT^{2} \)
71 \( 1 + (-7.03 + 7.03i)T - 71iT^{2} \)
73 \( 1 + 8.69iT - 73T^{2} \)
79 \( 1 + (8.32 - 8.32i)T - 79iT^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 + (6.59 - 6.59i)T - 89iT^{2} \)
97 \( 1 + (6.38 + 6.38i)T + 97iT^{2} \)
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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

−9.865359765351052217461146179498, −9.385144843685523432859201376012, −8.308513222227936142444122937402, −7.62660646640029583168251449323, −7.04786734455120780338230582266, −5.54458732151512525639948303962, −4.67127478132509410502709343412, −4.50489118672386590122645445368, −2.51644389984471113853586241900, −1.72022630961801231231410117797, 0.24991620282011700231987933335, 2.46771220789520746865260829560, 3.03432612172499560035867563937, 4.13867454139318501276669694843, 5.51532082485696561786791435320, 6.07945781836348405731720821176, 7.07208733118708939142214192290, 7.937376268581940773426816590013, 8.564722521305300886657252584516, 9.800368565666904087751406977551

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