Properties

Label 2-51e2-1.1-c1-0-91
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.43·2-s + 0.0711·4-s − 2.31·5-s + 4.44·7-s − 2.77·8-s − 3.33·10-s − 4.52·11-s + 1.17·13-s + 6.39·14-s − 4.13·16-s + 4.86·19-s − 0.164·20-s − 6.51·22-s − 0.625·23-s + 0.354·25-s + 1.69·26-s + 0.316·28-s − 1.51·29-s − 8.73·31-s − 0.402·32-s − 10.2·35-s − 8.79·37-s + 7.00·38-s + 6.42·40-s − 0.464·41-s − 1.51·43-s − 0.321·44-s + ⋯
L(s)  = 1  + 1.01·2-s + 0.0355·4-s − 1.03·5-s + 1.67·7-s − 0.981·8-s − 1.05·10-s − 1.36·11-s + 0.325·13-s + 1.70·14-s − 1.03·16-s + 1.11·19-s − 0.0368·20-s − 1.38·22-s − 0.130·23-s + 0.0708·25-s + 0.331·26-s + 0.0597·28-s − 0.280·29-s − 1.56·31-s − 0.0711·32-s − 1.73·35-s − 1.44·37-s + 1.13·38-s + 1.01·40-s − 0.0726·41-s − 0.230·43-s − 0.0485·44-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: \(1\)
Selberg data: \((2,\ 2601,\ (\ :1/2),\ -1)\)

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 \( 1 \)
17 \( 1 \)
good2 \( 1 - 1.43T + 2T^{2} \)
5 \( 1 + 2.31T + 5T^{2} \)
7 \( 1 - 4.44T + 7T^{2} \)
11 \( 1 + 4.52T + 11T^{2} \)
13 \( 1 - 1.17T + 13T^{2} \)
19 \( 1 - 4.86T + 19T^{2} \)
23 \( 1 + 0.625T + 23T^{2} \)
29 \( 1 + 1.51T + 29T^{2} \)
31 \( 1 + 8.73T + 31T^{2} \)
37 \( 1 + 8.79T + 37T^{2} \)
41 \( 1 + 0.464T + 41T^{2} \)
43 \( 1 + 1.51T + 43T^{2} \)
47 \( 1 + 6.01T + 47T^{2} \)
53 \( 1 + 7.12T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 + 2.55T + 61T^{2} \)
67 \( 1 + 3.71T + 67T^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 + 2.70T + 73T^{2} \)
79 \( 1 + 16.6T + 79T^{2} \)
83 \( 1 - 4.45T + 83T^{2} \)
89 \( 1 + 1.54T + 89T^{2} \)
97 \( 1 - 3.83T + 97T^{2} \)
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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

−8.329909440870071619393310299029, −7.73058024932352073324204305013, −7.14438105437553299074281542387, −5.68557989674245591880923341934, −5.23431876116795500506905933343, −4.60308636331657089248095392796, −3.79088041201331679649847508804, −3.00252751878126545453657420363, −1.71094289357146942503382595001, 0, 1.71094289357146942503382595001, 3.00252751878126545453657420363, 3.79088041201331679649847508804, 4.60308636331657089248095392796, 5.23431876116795500506905933343, 5.68557989674245591880923341934, 7.14438105437553299074281542387, 7.73058024932352073324204305013, 8.329909440870071619393310299029

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