Properties

Label 2-1148-1.1-c1-0-2
Degree $2$
Conductor $1148$
Sign $1$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.10·3-s − 1.26·5-s + 7-s + 1.42·9-s − 3.91·11-s + 2.38·13-s + 2.65·15-s − 6.14·17-s + 5.49·19-s − 2.10·21-s − 0.104·23-s − 3.40·25-s + 3.30·27-s − 2.65·29-s + 3.26·31-s + 8.24·33-s − 1.26·35-s + 0.993·37-s − 5.02·39-s + 41-s + 8.14·43-s − 1.80·45-s + 1.90·47-s + 49-s + 12.9·51-s + 1.26·53-s + 4.94·55-s + ⋯
L(s)  = 1  − 1.21·3-s − 0.564·5-s + 0.377·7-s + 0.475·9-s − 1.18·11-s + 0.662·13-s + 0.685·15-s − 1.49·17-s + 1.25·19-s − 0.459·21-s − 0.0217·23-s − 0.681·25-s + 0.636·27-s − 0.493·29-s + 0.585·31-s + 1.43·33-s − 0.213·35-s + 0.163·37-s − 0.804·39-s + 0.156·41-s + 1.24·43-s − 0.268·45-s + 0.277·47-s + 0.142·49-s + 1.81·51-s + 0.173·53-s + 0.666·55-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.7393427161\)
\(L(\frac12)\) \(\approx\) \(0.7393427161\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
41 \( 1 - T \)
good3 \( 1 + 2.10T + 3T^{2} \)
5 \( 1 + 1.26T + 5T^{2} \)
11 \( 1 + 3.91T + 11T^{2} \)
13 \( 1 - 2.38T + 13T^{2} \)
17 \( 1 + 6.14T + 17T^{2} \)
19 \( 1 - 5.49T + 19T^{2} \)
23 \( 1 + 0.104T + 23T^{2} \)
29 \( 1 + 2.65T + 29T^{2} \)
31 \( 1 - 3.26T + 31T^{2} \)
37 \( 1 - 0.993T + 37T^{2} \)
43 \( 1 - 8.14T + 43T^{2} \)
47 \( 1 - 1.90T + 47T^{2} \)
53 \( 1 - 1.26T + 53T^{2} \)
59 \( 1 - 8.24T + 59T^{2} \)
61 \( 1 - 6.89T + 61T^{2} \)
67 \( 1 - 8.69T + 67T^{2} \)
71 \( 1 - 0.827T + 71T^{2} \)
73 \( 1 - 8.44T + 73T^{2} \)
79 \( 1 + 5.27T + 79T^{2} \)
83 \( 1 - 6.82T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 8.31T + 97T^{2} \)
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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

−10.03743348881119897463837394811, −8.931927076093442905808792828881, −8.045521918082607150810484047670, −7.30884903311372884310247430503, −6.32570083991577565569218899577, −5.50220736714163124625417901771, −4.82480297161082963954035503469, −3.80818646040089337276809451920, −2.39841129364469488624720642148, −0.67292091057836015321014101113, 0.67292091057836015321014101113, 2.39841129364469488624720642148, 3.80818646040089337276809451920, 4.82480297161082963954035503469, 5.50220736714163124625417901771, 6.32570083991577565569218899577, 7.30884903311372884310247430503, 8.045521918082607150810484047670, 8.931927076093442905808792828881, 10.03743348881119897463837394811

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