Properties

Label 2-6025-1.1-c1-0-316
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  − 1.98·2-s + 2.93·3-s + 1.94·4-s − 5.82·6-s − 3.87·7-s + 0.108·8-s + 5.60·9-s + 5.43·11-s + 5.70·12-s + 4.51·13-s + 7.70·14-s − 4.10·16-s − 3.56·17-s − 11.1·18-s − 4.95·19-s − 11.3·21-s − 10.7·22-s − 8.91·23-s + 0.319·24-s − 8.96·26-s + 7.64·27-s − 7.54·28-s + 2.56·29-s − 1.87·31-s + 7.93·32-s + 15.9·33-s + 7.08·34-s + ⋯
L(s)  = 1  − 1.40·2-s + 1.69·3-s + 0.972·4-s − 2.37·6-s − 1.46·7-s + 0.0385·8-s + 1.86·9-s + 1.63·11-s + 1.64·12-s + 1.25·13-s + 2.05·14-s − 1.02·16-s − 0.864·17-s − 2.62·18-s − 1.13·19-s − 2.48·21-s − 2.30·22-s − 1.85·23-s + 0.0652·24-s − 1.75·26-s + 1.47·27-s − 1.42·28-s + 0.475·29-s − 0.336·31-s + 1.40·32-s + 2.77·33-s + 1.21·34-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 \( 1 + 1.98T + 2T^{2} \)
3 \( 1 - 2.93T + 3T^{2} \)
7 \( 1 + 3.87T + 7T^{2} \)
11 \( 1 - 5.43T + 11T^{2} \)
13 \( 1 - 4.51T + 13T^{2} \)
17 \( 1 + 3.56T + 17T^{2} \)
19 \( 1 + 4.95T + 19T^{2} \)
23 \( 1 + 8.91T + 23T^{2} \)
29 \( 1 - 2.56T + 29T^{2} \)
31 \( 1 + 1.87T + 31T^{2} \)
37 \( 1 + 8.26T + 37T^{2} \)
41 \( 1 - 1.60T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 0.538T + 47T^{2} \)
53 \( 1 - 6.76T + 53T^{2} \)
59 \( 1 + 7.99T + 59T^{2} \)
61 \( 1 + 6.30T + 61T^{2} \)
67 \( 1 - 6.40T + 67T^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 + 8.98T + 73T^{2} \)
79 \( 1 + 5.39T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + 3.42T + 97T^{2} \)
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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

−8.139857121535444819743625379883, −7.17887683802580634514950587224, −6.54724067055149404929587488534, −6.21821269002215278370602250604, −4.25370842288032477481475297349, −3.87333571549238204721776260089, −3.12542083016795341578275885094, −2.03833055038624911345205946778, −1.49024352183833019477856745664, 0, 1.49024352183833019477856745664, 2.03833055038624911345205946778, 3.12542083016795341578275885094, 3.87333571549238204721776260089, 4.25370842288032477481475297349, 6.21821269002215278370602250604, 6.54724067055149404929587488534, 7.17887683802580634514950587224, 8.139857121535444819743625379883

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