Properties

Label 2-51e2-1.1-c1-0-34
Degree $2$
Conductor $2601$
Sign $1$
Analytic cond. $20.7690$
Root an. cond. $4.55731$
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  + 1.76·2-s + 1.12·4-s − 3.31·5-s + 4.87·7-s − 1.55·8-s − 5.86·10-s + 2.38·11-s − 1.83·13-s + 8.61·14-s − 4.98·16-s + 1.94·19-s − 3.72·20-s + 4.20·22-s + 7.17·23-s + 6.02·25-s − 3.24·26-s + 5.46·28-s − 6.02·29-s + 5.65·31-s − 5.70·32-s − 16.1·35-s + 1.57·37-s + 3.43·38-s + 5.15·40-s + 10.4·41-s + 5.75·43-s + 2.66·44-s + ⋯
L(s)  = 1  + 1.24·2-s + 0.560·4-s − 1.48·5-s + 1.84·7-s − 0.549·8-s − 1.85·10-s + 0.717·11-s − 0.509·13-s + 2.30·14-s − 1.24·16-s + 0.445·19-s − 0.831·20-s + 0.896·22-s + 1.49·23-s + 1.20·25-s − 0.636·26-s + 1.03·28-s − 1.11·29-s + 1.01·31-s − 1.00·32-s − 2.73·35-s + 0.258·37-s + 0.556·38-s + 0.815·40-s + 1.63·41-s + 0.878·43-s + 0.402·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(20.7690\)
Root analytic conductor: \(4.55731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2601,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.126680363\)
\(L(\frac12)\) \(\approx\) \(3.126680363\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 - 1.76T + 2T^{2} \)
5 \( 1 + 3.31T + 5T^{2} \)
7 \( 1 - 4.87T + 7T^{2} \)
11 \( 1 - 2.38T + 11T^{2} \)
13 \( 1 + 1.83T + 13T^{2} \)
19 \( 1 - 1.94T + 19T^{2} \)
23 \( 1 - 7.17T + 23T^{2} \)
29 \( 1 + 6.02T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 - 1.57T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 - 5.75T + 43T^{2} \)
47 \( 1 + 2.91T + 47T^{2} \)
53 \( 1 + 9.23T + 53T^{2} \)
59 \( 1 + 0.726T + 59T^{2} \)
61 \( 1 - 7.22T + 61T^{2} \)
67 \( 1 - 5.14T + 67T^{2} \)
71 \( 1 - 9.34T + 71T^{2} \)
73 \( 1 + 1.34T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 5.37T + 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 1.08T + 97T^{2} \)
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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

−8.755915843672045738337223301004, −7.85908274907782430721994859638, −7.47302288067649819748867085609, −6.51219930810827614141206410126, −5.34975008659740938325764634560, −4.79341658122801858219108989720, −4.23021736437954355064180205222, −3.53802888330195643524042102655, −2.44243810478537382315465210172, −0.969255112279974773142271910569, 0.969255112279974773142271910569, 2.44243810478537382315465210172, 3.53802888330195643524042102655, 4.23021736437954355064180205222, 4.79341658122801858219108989720, 5.34975008659740938325764634560, 6.51219930810827614141206410126, 7.47302288067649819748867085609, 7.85908274907782430721994859638, 8.755915843672045738337223301004

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