Properties

Label 2-6025-1.1-c1-0-189
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  − 1.96·2-s − 2.44·3-s + 1.86·4-s + 4.81·6-s − 2.28·7-s + 0.256·8-s + 2.98·9-s + 4.45·11-s − 4.57·12-s + 2.62·13-s + 4.48·14-s − 4.24·16-s + 5.62·17-s − 5.87·18-s − 4.92·19-s + 5.58·21-s − 8.76·22-s + 2.14·23-s − 0.626·24-s − 5.17·26-s + 0.0375·27-s − 4.26·28-s − 1.59·29-s − 3.29·31-s + 7.83·32-s − 10.9·33-s − 11.0·34-s + ⋯
L(s)  = 1  − 1.39·2-s − 1.41·3-s + 0.934·4-s + 1.96·6-s − 0.862·7-s + 0.0905·8-s + 0.994·9-s + 1.34·11-s − 1.32·12-s + 0.729·13-s + 1.19·14-s − 1.06·16-s + 1.36·17-s − 1.38·18-s − 1.13·19-s + 1.21·21-s − 1.86·22-s + 0.446·23-s − 0.127·24-s − 1.01·26-s + 0.00722·27-s − 0.806·28-s − 0.297·29-s − 0.591·31-s + 1.38·32-s − 1.89·33-s − 1.89·34-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 \( 1 + 1.96T + 2T^{2} \)
3 \( 1 + 2.44T + 3T^{2} \)
7 \( 1 + 2.28T + 7T^{2} \)
11 \( 1 - 4.45T + 11T^{2} \)
13 \( 1 - 2.62T + 13T^{2} \)
17 \( 1 - 5.62T + 17T^{2} \)
19 \( 1 + 4.92T + 19T^{2} \)
23 \( 1 - 2.14T + 23T^{2} \)
29 \( 1 + 1.59T + 29T^{2} \)
31 \( 1 + 3.29T + 31T^{2} \)
37 \( 1 - 3.32T + 37T^{2} \)
41 \( 1 - 4.94T + 41T^{2} \)
43 \( 1 + 5.48T + 43T^{2} \)
47 \( 1 + 9.60T + 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 + 2.35T + 59T^{2} \)
61 \( 1 - 7.22T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 - 3.03T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 + 7.36T + 79T^{2} \)
83 \( 1 + 1.30T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + 8.92T + 97T^{2} \)
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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

−7.71507867841316258558613758989, −6.96248247086946119971313828254, −6.31675506069900507183606254427, −6.02889378383302408775113531075, −4.95542361524039038950340801036, −4.08972215812174837885884178319, −3.20466038823142273359653699268, −1.67944639814537746434409656694, −0.975704156680067257571672378654, 0, 0.975704156680067257571672378654, 1.67944639814537746434409656694, 3.20466038823142273359653699268, 4.08972215812174837885884178319, 4.95542361524039038950340801036, 6.02889378383302408775113531075, 6.31675506069900507183606254427, 6.96248247086946119971313828254, 7.71507867841316258558613758989

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