Properties

Label 2-2151-1.1-c1-0-84
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.61·2-s + 4.82·4-s + 3.37·5-s + 1.51·7-s + 7.37·8-s + 8.82·10-s − 1.81·11-s − 5.19·13-s + 3.95·14-s + 9.61·16-s − 4.20·17-s + 5.78·19-s + 16.2·20-s − 4.73·22-s − 1.61·23-s + 6.40·25-s − 13.5·26-s + 7.30·28-s − 4.27·29-s + 5.20·31-s + 10.3·32-s − 10.9·34-s + 5.11·35-s + 5.82·37-s + 15.1·38-s + 24.9·40-s − 9.18·41-s + ⋯
L(s)  = 1  + 1.84·2-s + 2.41·4-s + 1.51·5-s + 0.572·7-s + 2.60·8-s + 2.78·10-s − 0.546·11-s − 1.44·13-s + 1.05·14-s + 2.40·16-s − 1.02·17-s + 1.32·19-s + 3.64·20-s − 1.00·22-s − 0.335·23-s + 1.28·25-s − 2.66·26-s + 1.38·28-s − 0.794·29-s + 0.934·31-s + 1.83·32-s − 1.88·34-s + 0.864·35-s + 0.957·37-s + 2.45·38-s + 3.93·40-s − 1.43·41-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.172430958\)
\(L(\frac12)\) \(\approx\) \(7.172430958\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61T + 2T^{2} \)
5 \( 1 - 3.37T + 5T^{2} \)
7 \( 1 - 1.51T + 7T^{2} \)
11 \( 1 + 1.81T + 11T^{2} \)
13 \( 1 + 5.19T + 13T^{2} \)
17 \( 1 + 4.20T + 17T^{2} \)
19 \( 1 - 5.78T + 19T^{2} \)
23 \( 1 + 1.61T + 23T^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
31 \( 1 - 5.20T + 31T^{2} \)
37 \( 1 - 5.82T + 37T^{2} \)
41 \( 1 + 9.18T + 41T^{2} \)
43 \( 1 + 7.90T + 43T^{2} \)
47 \( 1 - 5.88T + 47T^{2} \)
53 \( 1 - 2.70T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 4.77T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 - 8.95T + 71T^{2} \)
73 \( 1 - 15.8T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−9.329583837531040765890371908381, −7.969320417129422489559339396080, −7.17439647157543497568755143962, −6.42008810994161346051357122788, −5.66447313697541033026270078712, −5.00980317274408757036180131936, −4.60785354473917755699825322589, −3.20264816559689057511025512459, −2.39402245390826079550308114539, −1.75867375465645989322237892959, 1.75867375465645989322237892959, 2.39402245390826079550308114539, 3.20264816559689057511025512459, 4.60785354473917755699825322589, 5.00980317274408757036180131936, 5.66447313697541033026270078712, 6.42008810994161346051357122788, 7.17439647157543497568755143962, 7.969320417129422489559339396080, 9.329583837531040765890371908381

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