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  − 0.277·2-s + 0.494·3-s − 1.92·4-s − 0.137·6-s − 1.36·7-s + 1.08·8-s − 2.75·9-s − 4.69·11-s − 0.951·12-s + 0.0431·13-s + 0.379·14-s + 3.54·16-s + 7.31·17-s + 0.764·18-s − 0.697·19-s − 0.676·21-s + 1.30·22-s − 1.41·23-s + 0.538·24-s − 0.0119·26-s − 2.84·27-s + 2.62·28-s + 8.30·29-s + 3.39·31-s − 3.16·32-s − 2.32·33-s − 2.03·34-s + ⋯
L(s)  = 1  − 0.196·2-s + 0.285·3-s − 0.961·4-s − 0.0560·6-s − 0.516·7-s + 0.384·8-s − 0.918·9-s − 1.41·11-s − 0.274·12-s + 0.0119·13-s + 0.101·14-s + 0.885·16-s + 1.77·17-s + 0.180·18-s − 0.160·19-s − 0.147·21-s + 0.278·22-s − 0.294·23-s + 0.109·24-s − 0.00235·26-s − 0.548·27-s + 0.496·28-s + 1.54·29-s + 0.610·31-s − 0.558·32-s − 0.404·33-s − 0.348·34-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 \( 1 + 0.277T + 2T^{2} \)
3 \( 1 - 0.494T + 3T^{2} \)
7 \( 1 + 1.36T + 7T^{2} \)
11 \( 1 + 4.69T + 11T^{2} \)
13 \( 1 - 0.0431T + 13T^{2} \)
17 \( 1 - 7.31T + 17T^{2} \)
19 \( 1 + 0.697T + 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 - 8.30T + 29T^{2} \)
31 \( 1 - 3.39T + 31T^{2} \)
37 \( 1 + 7.15T + 37T^{2} \)
41 \( 1 - 5.45T + 41T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 - 5.24T + 47T^{2} \)
53 \( 1 - 8.57T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 - 1.86T + 71T^{2} \)
73 \( 1 + 6.47T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 - 2.32T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 - 2.23T + 97T^{2} \)
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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

−7.85942955067612376354574871959, −7.33255710694547435283682405000, −6.03403373703361786706315474934, −5.60635024510859769152449243844, −4.90624483489919446719687390492, −4.00124615383734655925149610316, −3.09237554360249367464378793240, −2.61751433076654176149279470924, −1.05555259906198372615003846735, 0, 1.05555259906198372615003846735, 2.61751433076654176149279470924, 3.09237554360249367464378793240, 4.00124615383734655925149610316, 4.90624483489919446719687390492, 5.60635024510859769152449243844, 6.03403373703361786706315474934, 7.33255710694547435283682405000, 7.85942955067612376354574871959

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