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  + 2.49·2-s + 1.66·3-s + 4.22·4-s + 4.15·6-s − 4.24·7-s + 5.56·8-s − 0.229·9-s − 6.39·11-s + 7.03·12-s − 1.34·13-s − 10.6·14-s + 5.42·16-s + 1.79·17-s − 0.573·18-s − 3.80·19-s − 7.07·21-s − 15.9·22-s + 5.33·23-s + 9.25·24-s − 3.34·26-s − 5.37·27-s − 17.9·28-s + 1.62·29-s − 3.96·31-s + 2.41·32-s − 10.6·33-s + 4.48·34-s + ⋯
L(s)  = 1  + 1.76·2-s + 0.960·3-s + 2.11·4-s + 1.69·6-s − 1.60·7-s + 1.96·8-s − 0.0766·9-s − 1.92·11-s + 2.03·12-s − 0.371·13-s − 2.83·14-s + 1.35·16-s + 0.436·17-s − 0.135·18-s − 0.872·19-s − 1.54·21-s − 3.40·22-s + 1.11·23-s + 1.88·24-s − 0.656·26-s − 1.03·27-s − 3.39·28-s + 0.301·29-s − 0.712·31-s + 0.426·32-s − 1.85·33-s + 0.769·34-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 \( 1 - 2.49T + 2T^{2} \)
3 \( 1 - 1.66T + 3T^{2} \)
7 \( 1 + 4.24T + 7T^{2} \)
11 \( 1 + 6.39T + 11T^{2} \)
13 \( 1 + 1.34T + 13T^{2} \)
17 \( 1 - 1.79T + 17T^{2} \)
19 \( 1 + 3.80T + 19T^{2} \)
23 \( 1 - 5.33T + 23T^{2} \)
29 \( 1 - 1.62T + 29T^{2} \)
31 \( 1 + 3.96T + 31T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 - 2.25T + 41T^{2} \)
43 \( 1 - 1.94T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 - 9.06T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 + 4.44T + 67T^{2} \)
71 \( 1 + 6.47T + 71T^{2} \)
73 \( 1 + 4.44T + 73T^{2} \)
79 \( 1 + 9.81T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 - 5.74T + 97T^{2} \)
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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

−7.30742529015895188816467106239, −7.04493314420361343796724067431, −6.05881281319895162451138988414, −5.47826204380946018494336199510, −4.87339097122353380553390160553, −3.78515843160698395804871406610, −3.24594255189450890586038361703, −2.71988423154295930923540106378, −2.17989385199709490022857553406, 0, 2.17989385199709490022857553406, 2.71988423154295930923540106378, 3.24594255189450890586038361703, 3.78515843160698395804871406610, 4.87339097122353380553390160553, 5.47826204380946018494336199510, 6.05881281319895162451138988414, 7.04493314420361343796724067431, 7.30742529015895188816467106239

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