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  − 1.83·2-s − 2.40·3-s + 1.36·4-s + 4.41·6-s − 0.302·7-s + 1.17·8-s + 2.78·9-s − 5.09·11-s − 3.27·12-s + 4.63·13-s + 0.554·14-s − 4.87·16-s − 1.03·17-s − 5.11·18-s − 8.55·19-s + 0.727·21-s + 9.34·22-s + 1.34·23-s − 2.82·24-s − 8.49·26-s + 0.505·27-s − 0.411·28-s + 10.3·29-s + 0.459·31-s + 6.58·32-s + 12.2·33-s + 1.89·34-s + ⋯
L(s)  = 1  − 1.29·2-s − 1.38·3-s + 0.680·4-s + 1.80·6-s − 0.114·7-s + 0.414·8-s + 0.929·9-s − 1.53·11-s − 0.944·12-s + 1.28·13-s + 0.148·14-s − 1.21·16-s − 0.250·17-s − 1.20·18-s − 1.96·19-s + 0.158·21-s + 1.99·22-s + 0.280·23-s − 0.576·24-s − 1.66·26-s + 0.0973·27-s − 0.0777·28-s + 1.91·29-s + 0.0824·31-s + 1.16·32-s + 2.13·33-s + 0.325·34-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 \( 1 + 1.83T + 2T^{2} \)
3 \( 1 + 2.40T + 3T^{2} \)
7 \( 1 + 0.302T + 7T^{2} \)
11 \( 1 + 5.09T + 11T^{2} \)
13 \( 1 - 4.63T + 13T^{2} \)
17 \( 1 + 1.03T + 17T^{2} \)
19 \( 1 + 8.55T + 19T^{2} \)
23 \( 1 - 1.34T + 23T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 - 0.459T + 31T^{2} \)
37 \( 1 + 1.55T + 37T^{2} \)
41 \( 1 - 5.44T + 41T^{2} \)
43 \( 1 - 5.45T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 4.20T + 53T^{2} \)
59 \( 1 + 5.01T + 59T^{2} \)
61 \( 1 + 6.46T + 61T^{2} \)
67 \( 1 + 14.8T + 67T^{2} \)
71 \( 1 + 6.08T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 2.63T + 79T^{2} \)
83 \( 1 + 2.13T + 83T^{2} \)
89 \( 1 + 2.74T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−7.959145458462155008693716965033, −6.98099005833110392092845989091, −6.34007900159349055903265720488, −5.87323576294874323257296004164, −4.76770747573288234301835572655, −4.45154792792743030602129946050, −2.98316657133182769550058587410, −1.91509376511097174600562491451, −0.826641255013321944426850712358, 0, 0.826641255013321944426850712358, 1.91509376511097174600562491451, 2.98316657133182769550058587410, 4.45154792792743030602129946050, 4.76770747573288234301835572655, 5.87323576294874323257296004164, 6.34007900159349055903265720488, 6.98099005833110392092845989091, 7.959145458462155008693716965033

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