Properties

Label 2-2151-1.1-c1-0-37
Degree $2$
Conductor $2151$
Sign $-1$
Analytic cond. $17.1758$
Root an. cond. $4.14437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.05·2-s − 0.894·4-s − 2.87·5-s − 4.11·7-s + 3.04·8-s + 3.02·10-s + 1.17·11-s + 0.346·13-s + 4.32·14-s − 1.41·16-s + 3.26·17-s − 1.31·19-s + 2.56·20-s − 1.24·22-s + 9.10·23-s + 3.25·25-s − 0.364·26-s + 3.67·28-s + 0.639·29-s − 1.52·31-s − 4.60·32-s − 3.43·34-s + 11.8·35-s + 1.40·37-s + 1.38·38-s − 8.74·40-s + 2.17·41-s + ⋯
L(s)  = 1  − 0.743·2-s − 0.447·4-s − 1.28·5-s − 1.55·7-s + 1.07·8-s + 0.955·10-s + 0.355·11-s + 0.0960·13-s + 1.15·14-s − 0.353·16-s + 0.791·17-s − 0.302·19-s + 0.574·20-s − 0.264·22-s + 1.89·23-s + 0.650·25-s − 0.0714·26-s + 0.695·28-s + 0.118·29-s − 0.274·31-s − 0.813·32-s − 0.588·34-s + 1.99·35-s + 0.231·37-s + 0.225·38-s − 1.38·40-s + 0.339·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $-1$
Analytic conductor: \(17.1758\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 - T \)
good2 \( 1 + 1.05T + 2T^{2} \)
5 \( 1 + 2.87T + 5T^{2} \)
7 \( 1 + 4.11T + 7T^{2} \)
11 \( 1 - 1.17T + 11T^{2} \)
13 \( 1 - 0.346T + 13T^{2} \)
17 \( 1 - 3.26T + 17T^{2} \)
19 \( 1 + 1.31T + 19T^{2} \)
23 \( 1 - 9.10T + 23T^{2} \)
29 \( 1 - 0.639T + 29T^{2} \)
31 \( 1 + 1.52T + 31T^{2} \)
37 \( 1 - 1.40T + 37T^{2} \)
41 \( 1 - 2.17T + 41T^{2} \)
43 \( 1 + 1.97T + 43T^{2} \)
47 \( 1 - 1.69T + 47T^{2} \)
53 \( 1 - 0.0493T + 53T^{2} \)
59 \( 1 + 0.806T + 59T^{2} \)
61 \( 1 + 1.25T + 61T^{2} \)
67 \( 1 - 4.01T + 67T^{2} \)
71 \( 1 + 0.361T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 - 4.27T + 79T^{2} \)
83 \( 1 + 14.0T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−8.823551839313660128058415371815, −7.988704420625805837686966774494, −7.26155988837478946948405094011, −6.66990928397372597442512548421, −5.51635816153041091386451572408, −4.44015031778319853542313118208, −3.68589512581740716016483647849, −2.96991051792693047461880823104, −1.05092594542681997969987050282, 0, 1.05092594542681997969987050282, 2.96991051792693047461880823104, 3.68589512581740716016483647849, 4.44015031778319853542313118208, 5.51635816153041091386451572408, 6.66990928397372597442512548421, 7.26155988837478946948405094011, 7.988704420625805837686966774494, 8.823551839313660128058415371815

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