Properties

Label 2-55e2-1.1-c1-0-140
Degree $2$
Conductor $3025$
Sign $-1$
Analytic cond. $24.1547$
Root an. cond. $4.91474$
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  + 2.60·2-s − 2.79·3-s + 4.79·4-s − 7.27·6-s − 2.60·7-s + 7.27·8-s + 4.79·9-s − 13.3·12-s + 2.06·13-s − 6.79·14-s + 9.37·16-s − 4.66·17-s + 12.4·18-s − 7.27·19-s + 7.27·21-s − 2.20·23-s − 20.3·24-s + 5.37·26-s − 4.99·27-s − 12.4·28-s + 5.75·29-s − 5·31-s + 9.88·32-s − 12.1·34-s + 22.9·36-s + 0.582·37-s − 18.9·38-s + ⋯
L(s)  = 1  + 1.84·2-s − 1.61·3-s + 2.39·4-s − 2.96·6-s − 0.984·7-s + 2.57·8-s + 1.59·9-s − 3.86·12-s + 0.571·13-s − 1.81·14-s + 2.34·16-s − 1.13·17-s + 2.94·18-s − 1.66·19-s + 1.58·21-s − 0.460·23-s − 4.14·24-s + 1.05·26-s − 0.962·27-s − 2.35·28-s + 1.06·29-s − 0.898·31-s + 1.74·32-s − 2.08·34-s + 3.82·36-s + 0.0957·37-s − 3.07·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(24.1547\)
Root analytic conductor: \(4.91474\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3025,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 - 2.60T + 2T^{2} \)
3 \( 1 + 2.79T + 3T^{2} \)
7 \( 1 + 2.60T + 7T^{2} \)
13 \( 1 - 2.06T + 13T^{2} \)
17 \( 1 + 4.66T + 17T^{2} \)
19 \( 1 + 7.27T + 19T^{2} \)
23 \( 1 + 2.20T + 23T^{2} \)
29 \( 1 - 5.75T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 - 0.582T + 37T^{2} \)
41 \( 1 - 7.27T + 41T^{2} \)
43 \( 1 + 5.21T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 2.20T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 - 1.51T + 61T^{2} \)
67 \( 1 + 5.58T + 67T^{2} \)
71 \( 1 - 9.16T + 71T^{2} \)
73 \( 1 + 6.73T + 73T^{2} \)
79 \( 1 - 8.79T + 79T^{2} \)
83 \( 1 + 0.543T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−8.022119788408962787194667565295, −6.71506542037930826598458549234, −6.55221673002952835982412212628, −6.08781523983833724030976276601, −5.26032994689890223889942001943, −4.48804146063059028340765631951, −3.97300480364838044049344998481, −2.89593804408934521839801885803, −1.75629968365430245147695776899, 0, 1.75629968365430245147695776899, 2.89593804408934521839801885803, 3.97300480364838044049344998481, 4.48804146063059028340765631951, 5.26032994689890223889942001943, 6.08781523983833724030976276601, 6.55221673002952835982412212628, 6.71506542037930826598458549234, 8.022119788408962787194667565295

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