Properties

Label 2-6025-1.1-c1-0-100
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.409·2-s − 1.91·3-s − 1.83·4-s + 0.785·6-s − 3.66·7-s + 1.56·8-s + 0.682·9-s + 5.60·11-s + 3.51·12-s + 4.56·13-s + 1.49·14-s + 3.02·16-s + 6.06·17-s − 0.279·18-s − 3.67·19-s + 7.02·21-s − 2.29·22-s + 7.36·23-s − 3.01·24-s − 1.86·26-s + 4.44·27-s + 6.70·28-s − 1.37·29-s + 2.62·31-s − 4.37·32-s − 10.7·33-s − 2.48·34-s + ⋯
L(s)  = 1  − 0.289·2-s − 1.10·3-s − 0.916·4-s + 0.320·6-s − 1.38·7-s + 0.554·8-s + 0.227·9-s + 1.68·11-s + 1.01·12-s + 1.26·13-s + 0.400·14-s + 0.755·16-s + 1.47·17-s − 0.0658·18-s − 0.843·19-s + 1.53·21-s − 0.489·22-s + 1.53·23-s − 0.614·24-s − 0.366·26-s + 0.855·27-s + 1.26·28-s − 0.255·29-s + 0.471·31-s − 0.773·32-s − 1.87·33-s − 0.425·34-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: \(0\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9358942157\)
\(L(\frac12)\) \(\approx\) \(0.9358942157\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.409T + 2T^{2} \)
3 \( 1 + 1.91T + 3T^{2} \)
7 \( 1 + 3.66T + 7T^{2} \)
11 \( 1 - 5.60T + 11T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 - 6.06T + 17T^{2} \)
19 \( 1 + 3.67T + 19T^{2} \)
23 \( 1 - 7.36T + 23T^{2} \)
29 \( 1 + 1.37T + 29T^{2} \)
31 \( 1 - 2.62T + 31T^{2} \)
37 \( 1 - 3.20T + 37T^{2} \)
41 \( 1 + 9.05T + 41T^{2} \)
43 \( 1 - 8.36T + 43T^{2} \)
47 \( 1 - 8.91T + 47T^{2} \)
53 \( 1 - 8.02T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + 6.04T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 16.3T + 73T^{2} \)
79 \( 1 + 4.63T + 79T^{2} \)
83 \( 1 - 15.9T + 83T^{2} \)
89 \( 1 + 0.0114T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−8.295483515141071351151781236980, −7.07946330191002580724190371975, −6.62729040649800079556546825594, −5.83892660322075353607732235786, −5.52353537340499901978460362840, −4.33051948001064749362387464871, −3.79056689654946843695218767569, −3.05282220071781270040170741404, −1.20425653825144113952383106775, −0.70135469423396928413827311001, 0.70135469423396928413827311001, 1.20425653825144113952383106775, 3.05282220071781270040170741404, 3.79056689654946843695218767569, 4.33051948001064749362387464871, 5.52353537340499901978460362840, 5.83892660322075353607732235786, 6.62729040649800079556546825594, 7.07946330191002580724190371975, 8.295483515141071351151781236980

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