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Properties

Label 2-1148-41.18-c1-0-10
Degree $2$
Conductor $1148$
Sign $0.755 + 0.655i$
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  − 0.744·3-s + (−1.95 + 1.41i)5-s + (0.309 − 0.951i)7-s − 2.44·9-s + (0.733 + 0.532i)11-s + (0.410 + 1.26i)13-s + (1.45 − 1.05i)15-s + (−1.87 − 1.36i)17-s + (2.10 − 6.47i)19-s + (−0.229 + 0.707i)21-s + (−0.125 − 0.387i)23-s + (0.255 − 0.787i)25-s + 4.05·27-s + (−2.89 + 2.10i)29-s + (6.04 + 4.39i)31-s + ⋯
L(s)  = 1  − 0.429·3-s + (−0.873 + 0.634i)5-s + (0.116 − 0.359i)7-s − 0.815·9-s + (0.221 + 0.160i)11-s + (0.113 + 0.350i)13-s + (0.375 − 0.272i)15-s + (−0.454 − 0.329i)17-s + (0.482 − 1.48i)19-s + (−0.0501 + 0.154i)21-s + (−0.0262 − 0.0808i)23-s + (0.0511 − 0.157i)25-s + 0.779·27-s + (−0.537 + 0.390i)29-s + (1.08 + 0.789i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9084531775\)
\(L(\frac12)\) \(\approx\) \(0.9084531775\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-3.06 + 5.62i)T \)
good3 \( 1 + 0.744T + 3T^{2} \)
5 \( 1 + (1.95 - 1.41i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (-0.733 - 0.532i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.410 - 1.26i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.87 + 1.36i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.10 + 6.47i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.125 + 0.387i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.89 - 2.10i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-6.04 - 4.39i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-9.41 + 6.84i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-2.06 - 6.34i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (3.15 + 9.70i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.86 + 4.26i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.69 + 5.20i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.33 + 10.2i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (3.04 - 2.21i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-5.60 - 4.06i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 - 6.28T + 73T^{2} \)
79 \( 1 - 6.57T + 79T^{2} \)
83 \( 1 + 5.14T + 83T^{2} \)
89 \( 1 + (2.98 - 9.17i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.609 + 0.442i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.687959972120672036358301004041, −8.883840805348045114227474479164, −7.990019361561586795604001131052, −7.07665946476527537650422819111, −6.59445559671604197709071243268, −5.39394524294608971299966887806, −4.49895432655586547787214551087, −3.49602915746879102966148250059, −2.48744168446410619072041973059, −0.54592045003988424161862029109, 0.981991061905075289455924592806, 2.66832498128976623753852295138, 3.87060505925062891708548916880, 4.68826823216343249895945175427, 5.79394945828385845738443353977, 6.24567694222389100496927863904, 7.78330641219695903235834525832, 8.114189709937545940029869517180, 8.957398115134629012226956517960, 9.878202036671640006425410182861

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