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  + 1.35·2-s + 2.45·3-s − 0.160·4-s + 3.32·6-s + 0.283·7-s − 2.93·8-s + 3.00·9-s − 4.12·11-s − 0.393·12-s − 0.0271·13-s + 0.384·14-s − 3.65·16-s + 1.28·17-s + 4.07·18-s − 5.72·19-s + 0.695·21-s − 5.59·22-s + 5.97·23-s − 7.18·24-s − 0.0368·26-s + 0.0132·27-s − 0.0455·28-s − 2.55·29-s − 2.02·31-s + 0.905·32-s − 10.1·33-s + 1.74·34-s + ⋯
L(s)  = 1  + 0.959·2-s + 1.41·3-s − 0.0802·4-s + 1.35·6-s + 0.107·7-s − 1.03·8-s + 1.00·9-s − 1.24·11-s − 0.113·12-s − 0.00754·13-s + 0.102·14-s − 0.913·16-s + 0.312·17-s + 0.960·18-s − 1.31·19-s + 0.151·21-s − 1.19·22-s + 1.24·23-s − 1.46·24-s − 0.00723·26-s + 0.00254·27-s − 0.00860·28-s − 0.474·29-s − 0.364·31-s + 0.160·32-s − 1.76·33-s + 0.299·34-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 \( 1 - 1.35T + 2T^{2} \)
3 \( 1 - 2.45T + 3T^{2} \)
7 \( 1 - 0.283T + 7T^{2} \)
11 \( 1 + 4.12T + 11T^{2} \)
13 \( 1 + 0.0271T + 13T^{2} \)
17 \( 1 - 1.28T + 17T^{2} \)
19 \( 1 + 5.72T + 19T^{2} \)
23 \( 1 - 5.97T + 23T^{2} \)
29 \( 1 + 2.55T + 29T^{2} \)
31 \( 1 + 2.02T + 31T^{2} \)
37 \( 1 + 2.42T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + 4.54T + 47T^{2} \)
53 \( 1 - 9.30T + 53T^{2} \)
59 \( 1 + 9.94T + 59T^{2} \)
61 \( 1 - 8.17T + 61T^{2} \)
67 \( 1 + 4.40T + 67T^{2} \)
71 \( 1 + 3.80T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 - 6.69T + 79T^{2} \)
83 \( 1 - 4.32T + 83T^{2} \)
89 \( 1 - 0.746T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−7.892626039438091433523325396909, −7.02139303973405310264522688554, −6.27190439024025870943369858879, −5.15506161938530671803457351310, −4.93741155083991871791601314896, −3.81981217333182225157460398968, −3.32237164667051283804748097100, −2.63420767720936017327235550520, −1.83045457401636296032668057343, 0, 1.83045457401636296032668057343, 2.63420767720936017327235550520, 3.32237164667051283804748097100, 3.81981217333182225157460398968, 4.93741155083991871791601314896, 5.15506161938530671803457351310, 6.27190439024025870943369858879, 7.02139303973405310264522688554, 7.892626039438091433523325396909

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