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Properties

Label 2-1148-41.37-c1-0-1
Degree $2$
Conductor $1148$
Sign $-0.962 - 0.272i$
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  + 1.04·3-s + (−0.980 + 3.01i)5-s + (0.809 + 0.587i)7-s − 1.91·9-s + (−0.720 − 2.21i)11-s + (−4.96 + 3.60i)13-s + (−1.02 + 3.14i)15-s + (−0.584 − 1.79i)17-s + (0.874 + 0.635i)19-s + (0.842 + 0.612i)21-s + (−0.0492 + 0.0357i)23-s + (−4.10 − 2.98i)25-s − 5.11·27-s + (−2.10 + 6.46i)29-s + (−2.72 − 8.38i)31-s + ⋯
L(s)  = 1  + 0.601·3-s + (−0.438 + 1.34i)5-s + (0.305 + 0.222i)7-s − 0.638·9-s + (−0.217 − 0.668i)11-s + (−1.37 + 1.00i)13-s + (−0.263 + 0.811i)15-s + (−0.141 − 0.436i)17-s + (0.200 + 0.145i)19-s + (0.183 + 0.133i)21-s + (−0.0102 + 0.00745i)23-s + (−0.821 − 0.596i)25-s − 0.985·27-s + (−0.390 + 1.20i)29-s + (−0.489 − 1.50i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.962 - 0.272i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.962 - 0.272i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7802705795\)
\(L(\frac12)\) \(\approx\) \(0.7802705795\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.446 + 6.38i)T \)
good3 \( 1 - 1.04T + 3T^{2} \)
5 \( 1 + (0.980 - 3.01i)T + (-4.04 - 2.93i)T^{2} \)
11 \( 1 + (0.720 + 2.21i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (4.96 - 3.60i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.584 + 1.79i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.874 - 0.635i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.0492 - 0.0357i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.10 - 6.46i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.72 + 8.38i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.72 - 11.4i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-5.49 + 3.99i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-1.60 + 1.16i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.92 - 5.91i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.52 + 2.56i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (8.06 + 5.85i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.38 - 7.33i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-3.81 - 11.7i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 9.13T + 73T^{2} \)
79 \( 1 + 7.10T + 79T^{2} \)
83 \( 1 - 2.63T + 83T^{2} \)
89 \( 1 + (-4.99 - 3.63i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.62 - 11.1i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.16813810773691895605326107740, −9.277019957899242284029360472192, −8.562811012422336680662687103366, −7.54084447998498623156179368837, −7.13412428755527163479350253167, −6.06513567477918799929044970978, −5.04481780462670323327071607286, −3.79419781522715638474999898424, −2.91310829762626528220236208056, −2.21256306763270184464477881433, 0.28860244704323867739412889540, 1.91195231517373308694612193768, 3.07719139901091909716518040008, 4.29968561388253387447264975352, 5.00828534153798466271321136498, 5.78904648826977483695389909204, 7.40101242574307305614170281179, 7.79175854890040105750202537207, 8.621599763936224425299606659237, 9.251156742449059050258281977912

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