Properties

Label 2-6025-1.1-c1-0-250
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  + 2.40·2-s + 2.30·3-s + 3.78·4-s + 5.54·6-s − 4.72·7-s + 4.29·8-s + 2.31·9-s + 1.32·11-s + 8.73·12-s + 2.94·13-s − 11.3·14-s + 2.76·16-s + 2.94·17-s + 5.57·18-s + 5.65·19-s − 10.9·21-s + 3.19·22-s + 5.25·23-s + 9.91·24-s + 7.08·26-s − 1.57·27-s − 17.9·28-s + 6.80·29-s + 4.16·31-s − 1.94·32-s + 3.06·33-s + 7.07·34-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.33·3-s + 1.89·4-s + 2.26·6-s − 1.78·7-s + 1.51·8-s + 0.773·9-s + 0.400·11-s + 2.52·12-s + 0.817·13-s − 3.03·14-s + 0.691·16-s + 0.713·17-s + 1.31·18-s + 1.29·19-s − 2.37·21-s + 0.681·22-s + 1.09·23-s + 2.02·24-s + 1.39·26-s − 0.302·27-s − 3.38·28-s + 1.26·29-s + 0.747·31-s − 0.342·32-s + 0.533·33-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: \(0\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.528410982\)
\(L(\frac12)\) \(\approx\) \(8.528410982\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.40T + 2T^{2} \)
3 \( 1 - 2.30T + 3T^{2} \)
7 \( 1 + 4.72T + 7T^{2} \)
11 \( 1 - 1.32T + 11T^{2} \)
13 \( 1 - 2.94T + 13T^{2} \)
17 \( 1 - 2.94T + 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 - 5.25T + 23T^{2} \)
29 \( 1 - 6.80T + 29T^{2} \)
31 \( 1 - 4.16T + 31T^{2} \)
37 \( 1 + 1.74T + 37T^{2} \)
41 \( 1 + 7.98T + 41T^{2} \)
43 \( 1 + 0.0704T + 43T^{2} \)
47 \( 1 - 3.91T + 47T^{2} \)
53 \( 1 - 8.86T + 53T^{2} \)
59 \( 1 + 4.11T + 59T^{2} \)
61 \( 1 + 1.61T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 - 3.02T + 73T^{2} \)
79 \( 1 + 7.08T + 79T^{2} \)
83 \( 1 + 1.44T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 - 6.93T + 97T^{2} \)
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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

−7.983365734596655472517139386458, −6.85039245550467516359866122174, −6.81609237104130705015245921751, −5.82481170374842289303016258070, −5.21147016425311850605856917245, −4.05429279527940505076624385600, −3.55661581808571410687041744337, −3.01182749422009893581361504645, −2.63617649782340316142072107027, −1.19411997200796410975732870013, 1.19411997200796410975732870013, 2.63617649782340316142072107027, 3.01182749422009893581361504645, 3.55661581808571410687041744337, 4.05429279527940505076624385600, 5.21147016425311850605856917245, 5.82481170374842289303016258070, 6.81609237104130705015245921751, 6.85039245550467516359866122174, 7.983365734596655472517139386458

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