Properties

Label 2-2151-1.1-c3-0-32
Degree $2$
Conductor $2151$
Sign $1$
Analytic cond. $126.913$
Root an. cond. $11.2655$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.29·2-s + 20.0·4-s − 6.13·5-s − 8.89·7-s − 63.8·8-s + 32.5·10-s − 46.1·11-s − 27.2·13-s + 47.1·14-s + 177.·16-s + 106.·17-s + 45.2·19-s − 123.·20-s + 244.·22-s − 110.·23-s − 87.3·25-s + 144.·26-s − 178.·28-s + 248.·29-s + 299.·31-s − 429.·32-s − 565.·34-s + 54.6·35-s − 67.4·37-s − 239.·38-s + 391.·40-s + 76.0·41-s + ⋯
L(s)  = 1  − 1.87·2-s + 2.50·4-s − 0.549·5-s − 0.480·7-s − 2.81·8-s + 1.02·10-s − 1.26·11-s − 0.582·13-s + 0.899·14-s + 2.77·16-s + 1.52·17-s + 0.546·19-s − 1.37·20-s + 2.36·22-s − 0.997·23-s − 0.698·25-s + 1.08·26-s − 1.20·28-s + 1.58·29-s + 1.73·31-s − 2.37·32-s − 2.85·34-s + 0.263·35-s − 0.299·37-s − 1.02·38-s + 1.54·40-s + 0.289·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $1$
Analytic conductor: \(126.913\)
Root analytic conductor: \(11.2655\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3180233814\)
\(L(\frac12)\) \(\approx\) \(0.3180233814\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 - 239T \)
good2 \( 1 + 5.29T + 8T^{2} \)
5 \( 1 + 6.13T + 125T^{2} \)
7 \( 1 + 8.89T + 343T^{2} \)
11 \( 1 + 46.1T + 1.33e3T^{2} \)
13 \( 1 + 27.2T + 2.19e3T^{2} \)
17 \( 1 - 106.T + 4.91e3T^{2} \)
19 \( 1 - 45.2T + 6.85e3T^{2} \)
23 \( 1 + 110.T + 1.21e4T^{2} \)
29 \( 1 - 248.T + 2.43e4T^{2} \)
31 \( 1 - 299.T + 2.97e4T^{2} \)
37 \( 1 + 67.4T + 5.06e4T^{2} \)
41 \( 1 - 76.0T + 6.89e4T^{2} \)
43 \( 1 + 377.T + 7.95e4T^{2} \)
47 \( 1 + 424.T + 1.03e5T^{2} \)
53 \( 1 + 148.T + 1.48e5T^{2} \)
59 \( 1 + 540.T + 2.05e5T^{2} \)
61 \( 1 - 454.T + 2.26e5T^{2} \)
67 \( 1 + 262.T + 3.00e5T^{2} \)
71 \( 1 + 504.T + 3.57e5T^{2} \)
73 \( 1 + 270.T + 3.89e5T^{2} \)
79 \( 1 + 1.30e3T + 4.93e5T^{2} \)
83 \( 1 - 1.15e3T + 5.71e5T^{2} \)
89 \( 1 + 648.T + 7.04e5T^{2} \)
97 \( 1 - 238.T + 9.12e5T^{2} \)
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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

−8.539431621178182620307345387954, −7.927236587129662552437173206315, −7.68737149041510461273661479857, −6.69493413494211742540581285815, −5.93321473851282262242853666787, −4.83811919323769103912515986074, −3.25412946737603249109679119599, −2.68712583868952698571661381156, −1.44518299461640324851108293548, −0.34920883187839432969538437933, 0.34920883187839432969538437933, 1.44518299461640324851108293548, 2.68712583868952698571661381156, 3.25412946737603249109679119599, 4.83811919323769103912515986074, 5.93321473851282262242853666787, 6.69493413494211742540581285815, 7.68737149041510461273661479857, 7.927236587129662552437173206315, 8.539431621178182620307345387954

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