Properties

Degree $2$
Conductor $6025$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.91·2-s + 0.186·3-s + 1.65·4-s + 0.355·6-s − 3.52·7-s − 0.663·8-s − 2.96·9-s + 0.515·11-s + 0.307·12-s + 5.38·13-s − 6.74·14-s − 4.57·16-s + 4.16·17-s − 5.66·18-s + 4.92·19-s − 0.657·21-s + 0.985·22-s + 7.69·23-s − 0.123·24-s + 10.2·26-s − 1.11·27-s − 5.83·28-s − 8.93·29-s − 4.43·31-s − 7.41·32-s + 0.0959·33-s + 7.96·34-s + ⋯
L(s)  = 1  + 1.35·2-s + 0.107·3-s + 0.826·4-s + 0.145·6-s − 1.33·7-s − 0.234·8-s − 0.988·9-s + 0.155·11-s + 0.0888·12-s + 1.49·13-s − 1.80·14-s − 1.14·16-s + 1.01·17-s − 1.33·18-s + 1.13·19-s − 0.143·21-s + 0.210·22-s + 1.60·23-s − 0.0252·24-s + 2.01·26-s − 0.213·27-s − 1.10·28-s − 1.65·29-s − 0.795·31-s − 1.31·32-s + 0.0167·33-s + 1.36·34-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{6025} (1, \cdot )$
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 \( 1 - 1.91T + 2T^{2} \)
3 \( 1 - 0.186T + 3T^{2} \)
7 \( 1 + 3.52T + 7T^{2} \)
11 \( 1 - 0.515T + 11T^{2} \)
13 \( 1 - 5.38T + 13T^{2} \)
17 \( 1 - 4.16T + 17T^{2} \)
19 \( 1 - 4.92T + 19T^{2} \)
23 \( 1 - 7.69T + 23T^{2} \)
29 \( 1 + 8.93T + 29T^{2} \)
31 \( 1 + 4.43T + 31T^{2} \)
37 \( 1 + 5.99T + 37T^{2} \)
41 \( 1 + 8.99T + 41T^{2} \)
43 \( 1 - 1.66T + 43T^{2} \)
47 \( 1 + 8.55T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 - 9.25T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 - 3.91T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + 1.43T + 79T^{2} \)
83 \( 1 + 1.73T + 83T^{2} \)
89 \( 1 + 1.07T + 89T^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
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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

−7.45781382945161607937413221660, −6.74558119704038801557956336352, −6.04742598004296656365346102303, −5.58453954017203287209056791790, −4.99125423344901208041356618688, −3.71447690082745378758782359501, −3.27782049858611290373735521444, −3.05791456243855914230237755041, −1.51514474603164630693546597119, 0, 1.51514474603164630693546597119, 3.05791456243855914230237755041, 3.27782049858611290373735521444, 3.71447690082745378758782359501, 4.99125423344901208041356618688, 5.58453954017203287209056791790, 6.04742598004296656365346102303, 6.74558119704038801557956336352, 7.45781382945161607937413221660

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