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  − 0.760·2-s + 1.12·3-s − 1.42·4-s − 0.859·6-s + 2.53·7-s + 2.60·8-s − 1.72·9-s − 2.06·11-s − 1.60·12-s − 4.79·13-s − 1.93·14-s + 0.862·16-s − 1.95·17-s + 1.31·18-s − 1.03·19-s + 2.86·21-s + 1.56·22-s + 2.68·23-s + 2.93·24-s + 3.64·26-s − 5.33·27-s − 3.60·28-s + 5.84·29-s + 7.76·31-s − 5.86·32-s − 2.32·33-s + 1.48·34-s + ⋯
L(s)  = 1  − 0.537·2-s + 0.652·3-s − 0.710·4-s − 0.350·6-s + 0.959·7-s + 0.920·8-s − 0.574·9-s − 0.621·11-s − 0.463·12-s − 1.33·13-s − 0.516·14-s + 0.215·16-s − 0.473·17-s + 0.309·18-s − 0.238·19-s + 0.625·21-s + 0.334·22-s + 0.560·23-s + 0.600·24-s + 0.715·26-s − 1.02·27-s − 0.681·28-s + 1.08·29-s + 1.39·31-s − 1.03·32-s − 0.405·33-s + 0.254·34-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 \( 1 + 0.760T + 2T^{2} \)
3 \( 1 - 1.12T + 3T^{2} \)
7 \( 1 - 2.53T + 7T^{2} \)
11 \( 1 + 2.06T + 11T^{2} \)
13 \( 1 + 4.79T + 13T^{2} \)
17 \( 1 + 1.95T + 17T^{2} \)
19 \( 1 + 1.03T + 19T^{2} \)
23 \( 1 - 2.68T + 23T^{2} \)
29 \( 1 - 5.84T + 29T^{2} \)
31 \( 1 - 7.76T + 31T^{2} \)
37 \( 1 - 9.08T + 37T^{2} \)
41 \( 1 - 9.18T + 41T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 + 4.22T + 47T^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 - 2.97T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 1.40T + 67T^{2} \)
71 \( 1 + 6.51T + 71T^{2} \)
73 \( 1 - 1.89T + 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 + 6.49T + 83T^{2} \)
89 \( 1 + 0.236T + 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−7.78422803829809533044685046794, −7.55545252859081014311971034434, −6.35802052308283880769739207688, −5.40181846285169934255768654809, −4.64123442358340373136005928987, −4.34849597422386875853838500377, −2.90483542429481885136159282256, −2.44220197767380794751746045498, −1.22914166614404440481363436720, 0, 1.22914166614404440481363436720, 2.44220197767380794751746045498, 2.90483542429481885136159282256, 4.34849597422386875853838500377, 4.64123442358340373136005928987, 5.40181846285169934255768654809, 6.35802052308283880769739207688, 7.55545252859081014311971034434, 7.78422803829809533044685046794

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