Properties

Degree 2
Conductor $ 5^{2} \cdot 241 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.60·2-s − 0.980·3-s + 4.77·4-s − 2.55·6-s + 1.30·7-s + 7.23·8-s − 2.03·9-s − 3.27·11-s − 4.68·12-s − 4.30·13-s + 3.39·14-s + 9.27·16-s + 1.02·17-s − 5.31·18-s − 7.01·19-s − 1.27·21-s − 8.51·22-s − 0.835·23-s − 7.09·24-s − 11.2·26-s + 4.93·27-s + 6.24·28-s − 1.11·29-s − 3.97·31-s + 9.69·32-s + 3.20·33-s + 2.66·34-s + ⋯
L(s)  = 1  + 1.84·2-s − 0.565·3-s + 2.38·4-s − 1.04·6-s + 0.493·7-s + 2.55·8-s − 0.679·9-s − 0.986·11-s − 1.35·12-s − 1.19·13-s + 0.908·14-s + 2.31·16-s + 0.248·17-s − 1.25·18-s − 1.60·19-s − 0.279·21-s − 1.81·22-s − 0.174·23-s − 1.44·24-s − 2.19·26-s + 0.950·27-s + 1.17·28-s − 0.207·29-s − 0.713·31-s + 1.71·32-s + 0.558·33-s + 0.457·34-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

\( d \)  =  \(2\)
\( N \)  =  \(6025\)    =    \(5^{2} \cdot 241\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6025} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 6025,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;241\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;241\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 + T \)
good2 \( 1 - 2.60T + 2T^{2} \)
3 \( 1 + 0.980T + 3T^{2} \)
7 \( 1 - 1.30T + 7T^{2} \)
11 \( 1 + 3.27T + 11T^{2} \)
13 \( 1 + 4.30T + 13T^{2} \)
17 \( 1 - 1.02T + 17T^{2} \)
19 \( 1 + 7.01T + 19T^{2} \)
23 \( 1 + 0.835T + 23T^{2} \)
29 \( 1 + 1.11T + 29T^{2} \)
31 \( 1 + 3.97T + 31T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 - 1.22T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 - 0.151T + 47T^{2} \)
53 \( 1 + 3.02T + 53T^{2} \)
59 \( 1 + 4.15T + 59T^{2} \)
61 \( 1 - 5.62T + 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 + 1.66T + 79T^{2} \)
83 \( 1 - 2.34T + 83T^{2} \)
89 \( 1 - 18.1T + 89T^{2} \)
97 \( 1 - 7.17T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.56049884832351607894931185561, −6.64905162949504566246166140926, −6.02837132733721585605600931703, −5.47586674925272142895011641924, −4.74742714976800851740862656130, −4.45385736709176350455570641449, −3.26537205715113307767772010714, −2.58066776394611202272409581501, −1.87012585891195728389189066642, 0, 1.87012585891195728389189066642, 2.58066776394611202272409581501, 3.26537205715113307767772010714, 4.45385736709176350455570641449, 4.74742714976800851740862656130, 5.47586674925272142895011641924, 6.02837132733721585605600931703, 6.64905162949504566246166140926, 7.56049884832351607894931185561

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