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  − 1.63·2-s + 1.16·3-s + 0.660·4-s − 1.90·6-s − 5.06·7-s + 2.18·8-s − 1.63·9-s + 1.08·11-s + 0.771·12-s − 3.01·13-s + 8.25·14-s − 4.88·16-s − 2.47·17-s + 2.66·18-s − 7.12·19-s − 5.91·21-s − 1.76·22-s − 5.33·23-s + 2.55·24-s + 4.91·26-s − 5.41·27-s − 3.34·28-s − 6.80·29-s − 8.37·31-s + 3.60·32-s + 1.26·33-s + 4.04·34-s + ⋯
L(s)  = 1  − 1.15·2-s + 0.674·3-s + 0.330·4-s − 0.777·6-s − 1.91·7-s + 0.772·8-s − 0.545·9-s + 0.325·11-s + 0.222·12-s − 0.835·13-s + 2.20·14-s − 1.22·16-s − 0.600·17-s + 0.629·18-s − 1.63·19-s − 1.28·21-s − 0.376·22-s − 1.11·23-s + 0.520·24-s + 0.963·26-s − 1.04·27-s − 0.632·28-s − 1.26·29-s − 1.50·31-s + 0.636·32-s + 0.219·33-s + 0.692·34-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  =  $(2,\ 6025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.02027754138$
$L(\frac12)$  $\approx$  $0.02027754138$
$L(\frac{3}{2})$   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 \( 1 + 1.63T + 2T^{2} \)
3 \( 1 - 1.16T + 3T^{2} \)
7 \( 1 + 5.06T + 7T^{2} \)
11 \( 1 - 1.08T + 11T^{2} \)
13 \( 1 + 3.01T + 13T^{2} \)
17 \( 1 + 2.47T + 17T^{2} \)
19 \( 1 + 7.12T + 19T^{2} \)
23 \( 1 + 5.33T + 23T^{2} \)
29 \( 1 + 6.80T + 29T^{2} \)
31 \( 1 + 8.37T + 31T^{2} \)
37 \( 1 - 2.09T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 - 5.49T + 43T^{2} \)
47 \( 1 + 4.90T + 47T^{2} \)
53 \( 1 - 4.04T + 53T^{2} \)
59 \( 1 - 5.64T + 59T^{2} \)
61 \( 1 + 2.03T + 61T^{2} \)
67 \( 1 + 7.65T + 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 + 0.733T + 73T^{2} \)
79 \( 1 - 6.86T + 79T^{2} \)
83 \( 1 + 5.58T + 83T^{2} \)
89 \( 1 - 9.62T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−8.246971944494751038916263673128, −7.53814345017532462503170356232, −6.81101737754490964204557505873, −6.26887347461698481049707693699, −5.35087149305653422606806739204, −4.07016131590709635732977205718, −3.63549266374328000183717492891, −2.50650366621683987213067879976, −1.95611075712301800360198119634, −0.079357232701144311765082403836, 0.079357232701144311765082403836, 1.95611075712301800360198119634, 2.50650366621683987213067879976, 3.63549266374328000183717492891, 4.07016131590709635732977205718, 5.35087149305653422606806739204, 6.26887347461698481049707693699, 6.81101737754490964204557505873, 7.53814345017532462503170356232, 8.246971944494751038916263673128

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