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Properties

Label 2-1148-41.32-c1-0-7
Degree $2$
Conductor $1148$
Sign $0.337 - 0.941i$
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  + (1.71 − 1.71i)3-s + 2.46i·5-s + (−0.707 + 0.707i)7-s − 2.85i·9-s + (−3.89 + 3.89i)11-s + (−2.36 + 2.36i)13-s + (4.22 + 4.22i)15-s + (1.41 + 1.41i)17-s + (2.77 + 2.77i)19-s + 2.42i·21-s − 6.33·23-s − 1.09·25-s + (0.244 + 0.244i)27-s + (−1.38 + 1.38i)29-s + 6.75·31-s + ⋯
L(s)  = 1  + (0.988 − 0.988i)3-s + 1.10i·5-s + (−0.267 + 0.267i)7-s − 0.952i·9-s + (−1.17 + 1.17i)11-s + (−0.654 + 0.654i)13-s + (1.09 + 1.09i)15-s + (0.344 + 0.344i)17-s + (0.635 + 0.635i)19-s + 0.528i·21-s − 1.32·23-s − 0.219·25-s + (0.0470 + 0.0470i)27-s + (−0.256 + 0.256i)29-s + 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.337 - 0.941i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.337 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.698220484\)
\(L(\frac12)\) \(\approx\) \(1.698220484\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.55 + 4.49i)T \)
good3 \( 1 + (-1.71 + 1.71i)T - 3iT^{2} \)
5 \( 1 - 2.46iT - 5T^{2} \)
11 \( 1 + (3.89 - 3.89i)T - 11iT^{2} \)
13 \( 1 + (2.36 - 2.36i)T - 13iT^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \)
19 \( 1 + (-2.77 - 2.77i)T + 19iT^{2} \)
23 \( 1 + 6.33T + 23T^{2} \)
29 \( 1 + (1.38 - 1.38i)T - 29iT^{2} \)
31 \( 1 - 6.75T + 31T^{2} \)
37 \( 1 - 0.731T + 37T^{2} \)
43 \( 1 - 7.12iT - 43T^{2} \)
47 \( 1 + (5.61 + 5.61i)T + 47iT^{2} \)
53 \( 1 + (-4.60 + 4.60i)T - 53iT^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 - 9.57iT - 61T^{2} \)
67 \( 1 + (3.13 + 3.13i)T + 67iT^{2} \)
71 \( 1 + (2.44 - 2.44i)T - 71iT^{2} \)
73 \( 1 + 13.6iT - 73T^{2} \)
79 \( 1 + (1.22 - 1.22i)T - 79iT^{2} \)
83 \( 1 - 6.98T + 83T^{2} \)
89 \( 1 + (-11.5 + 11.5i)T - 89iT^{2} \)
97 \( 1 + (-11.0 - 11.0i)T + 97iT^{2} \)
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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

−10.03888435728165638600392487757, −9.120792142086362236291197603536, −7.918306342008794460029186697988, −7.63423495908475593628994112973, −6.87198748636944717125003427374, −6.04203317866124966956892430350, −4.74783731695989142230730908524, −3.39230059045728618880132811982, −2.52959994068204849953040952603, −1.89635269311481446099248893466, 0.62200827317055214798882982822, 2.63113267452774179949638856583, 3.31528500302430706625628520797, 4.44977655109039681698448162495, 5.12093241748490206295577898066, 6.01842439572065138284085576455, 7.63021050516150560056025826503, 8.148172639042024623885308482546, 8.837938878371781617996367430066, 9.666776536649850513330671869321

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