Properties

Degree 2
Conductor $ 5^{2} \cdot 241 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.54·2-s − 2.81·3-s + 0.398·4-s + 4.35·6-s + 4.24·7-s + 2.47·8-s + 4.90·9-s + 0.915·11-s − 1.12·12-s − 4.81·13-s − 6.57·14-s − 4.63·16-s + 5.38·17-s − 7.59·18-s − 4.34·19-s − 11.9·21-s − 1.41·22-s − 8.10·23-s − 6.97·24-s + 7.45·26-s − 5.34·27-s + 1.69·28-s − 6.45·29-s + 10.7·31-s + 2.22·32-s − 2.57·33-s − 8.34·34-s + ⋯
L(s)  = 1  − 1.09·2-s − 1.62·3-s + 0.199·4-s + 1.77·6-s + 1.60·7-s + 0.876·8-s + 1.63·9-s + 0.276·11-s − 0.323·12-s − 1.33·13-s − 1.75·14-s − 1.15·16-s + 1.30·17-s − 1.78·18-s − 0.997·19-s − 2.60·21-s − 0.302·22-s − 1.69·23-s − 1.42·24-s + 1.46·26-s − 1.02·27-s + 0.319·28-s − 1.19·29-s + 1.93·31-s + 0.393·32-s − 0.447·33-s − 1.43·34-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  =  0
Selberg data  =  $(2,\ 6025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4905642059$
$L(\frac12)$  $\approx$  $0.4905642059$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 \)
241 \( 1 - T \)
good2 \( 1 + 1.54T + 2T^{2} \)
3 \( 1 + 2.81T + 3T^{2} \)
7 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 - 0.915T + 11T^{2} \)
13 \( 1 + 4.81T + 13T^{2} \)
17 \( 1 - 5.38T + 17T^{2} \)
19 \( 1 + 4.34T + 19T^{2} \)
23 \( 1 + 8.10T + 23T^{2} \)
29 \( 1 + 6.45T + 29T^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 + 5.16T + 37T^{2} \)
41 \( 1 + 0.612T + 41T^{2} \)
43 \( 1 - 1.85T + 43T^{2} \)
47 \( 1 - 2.21T + 47T^{2} \)
53 \( 1 + 0.00846T + 53T^{2} \)
59 \( 1 - 8.85T + 59T^{2} \)
61 \( 1 - 3.78T + 61T^{2} \)
67 \( 1 + 4.67T + 67T^{2} \)
71 \( 1 - 1.48T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 + 17.6T + 79T^{2} \)
83 \( 1 + 7.45T + 83T^{2} \)
89 \( 1 + 0.520T + 89T^{2} \)
97 \( 1 - 6.33T + 97T^{2} \)
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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

−8.040468046723424358027273412779, −7.50291384883029573159778799725, −6.83478544098632697730527105972, −5.84741551776883485929205840365, −5.26488126375784386549407764930, −4.59934491311818195272861161603, −4.09333825016662926515373073917, −2.18857653605444168935949099546, −1.45869755532248128003965703726, −0.51385270651360754836525678170, 0.51385270651360754836525678170, 1.45869755532248128003965703726, 2.18857653605444168935949099546, 4.09333825016662926515373073917, 4.59934491311818195272861161603, 5.26488126375784386549407764930, 5.84741551776883485929205840365, 6.83478544098632697730527105972, 7.50291384883029573159778799725, 8.040468046723424358027273412779

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