Properties

Label 2-6025-1.1-c1-0-202
Degree $2$
Conductor $6025$
Sign $-1$
Analytic cond. $48.1098$
Root an. cond. $6.93612$
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  − 1.92·2-s − 0.860·3-s + 1.72·4-s + 1.66·6-s + 4.27·7-s + 0.532·8-s − 2.25·9-s − 4.15·11-s − 1.48·12-s − 1.01·13-s − 8.25·14-s − 4.47·16-s − 5.08·17-s + 4.35·18-s − 0.226·19-s − 3.68·21-s + 8.02·22-s + 2.25·23-s − 0.458·24-s + 1.95·26-s + 4.52·27-s + 7.37·28-s − 0.0980·29-s + 5.03·31-s + 7.57·32-s + 3.58·33-s + 9.82·34-s + ⋯
L(s)  = 1  − 1.36·2-s − 0.497·3-s + 0.861·4-s + 0.678·6-s + 1.61·7-s + 0.188·8-s − 0.752·9-s − 1.25·11-s − 0.428·12-s − 0.281·13-s − 2.20·14-s − 1.11·16-s − 1.23·17-s + 1.02·18-s − 0.0520·19-s − 0.803·21-s + 1.71·22-s + 0.470·23-s − 0.0936·24-s + 0.383·26-s + 0.871·27-s + 1.39·28-s − 0.0182·29-s + 0.905·31-s + 1.33·32-s + 0.623·33-s + 1.68·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-1$
Analytic conductor: \(48.1098\)
Root analytic conductor: \(6.93612\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ -1)\)

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 \( 1 \)
241 \( 1 + T \)
good2 \( 1 + 1.92T + 2T^{2} \)
3 \( 1 + 0.860T + 3T^{2} \)
7 \( 1 - 4.27T + 7T^{2} \)
11 \( 1 + 4.15T + 11T^{2} \)
13 \( 1 + 1.01T + 13T^{2} \)
17 \( 1 + 5.08T + 17T^{2} \)
19 \( 1 + 0.226T + 19T^{2} \)
23 \( 1 - 2.25T + 23T^{2} \)
29 \( 1 + 0.0980T + 29T^{2} \)
31 \( 1 - 5.03T + 31T^{2} \)
37 \( 1 - 1.95T + 37T^{2} \)
41 \( 1 - 1.64T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + 9.73T + 47T^{2} \)
53 \( 1 - 8.87T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 5.46T + 61T^{2} \)
67 \( 1 - 5.19T + 67T^{2} \)
71 \( 1 - 2.82T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 6.82T + 79T^{2} \)
83 \( 1 + 0.598T + 83T^{2} \)
89 \( 1 - 4.03T + 89T^{2} \)
97 \( 1 - 6.64T + 97T^{2} \)
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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

−7.86428233335964302152816206642, −7.38235326076402630100355884943, −6.48128269817396910593663978679, −5.54975338966373957727888146654, −4.85119311591595047095230901794, −4.39911840414514454046899715130, −2.71381043846811850640646854312, −2.12457994108672616023204691472, −1.03154200187763773647899555819, 0, 1.03154200187763773647899555819, 2.12457994108672616023204691472, 2.71381043846811850640646854312, 4.39911840414514454046899715130, 4.85119311591595047095230901794, 5.54975338966373957727888146654, 6.48128269817396910593663978679, 7.38235326076402630100355884943, 7.86428233335964302152816206642

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