Properties

Label 2-75e2-1.1-c1-0-30
Degree $2$
Conductor $5625$
Sign $1$
Analytic cond. $44.9158$
Root an. cond. $6.70192$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.54·2-s + 0.373·4-s − 3.37·7-s − 2.50·8-s − 0.275·11-s − 5.74·13-s − 5.20·14-s − 4.60·16-s + 6.11·17-s − 5.04·19-s − 0.424·22-s + 2.66·23-s − 8.84·26-s − 1.26·28-s + 6.27·29-s + 0.987·31-s − 2.08·32-s + 9.42·34-s − 6.20·37-s − 7.77·38-s + 0.876·41-s − 0.0269·43-s − 0.103·44-s + 4.10·46-s + 11.2·47-s + 4.39·49-s − 2.14·52-s + ⋯
L(s)  = 1  + 1.08·2-s + 0.186·4-s − 1.27·7-s − 0.885·8-s − 0.0830·11-s − 1.59·13-s − 1.39·14-s − 1.15·16-s + 1.48·17-s − 1.15·19-s − 0.0904·22-s + 0.555·23-s − 1.73·26-s − 0.238·28-s + 1.16·29-s + 0.177·31-s − 0.369·32-s + 1.61·34-s − 1.02·37-s − 1.26·38-s + 0.136·41-s − 0.00411·43-s − 0.0155·44-s + 0.605·46-s + 1.64·47-s + 0.628·49-s − 0.297·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(44.9158\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5625,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.788635623\)
\(L(\frac12)\) \(\approx\) \(1.788635623\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 1.54T + 2T^{2} \)
7 \( 1 + 3.37T + 7T^{2} \)
11 \( 1 + 0.275T + 11T^{2} \)
13 \( 1 + 5.74T + 13T^{2} \)
17 \( 1 - 6.11T + 17T^{2} \)
19 \( 1 + 5.04T + 19T^{2} \)
23 \( 1 - 2.66T + 23T^{2} \)
29 \( 1 - 6.27T + 29T^{2} \)
31 \( 1 - 0.987T + 31T^{2} \)
37 \( 1 + 6.20T + 37T^{2} \)
41 \( 1 - 0.876T + 41T^{2} \)
43 \( 1 + 0.0269T + 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 + 2.16T + 53T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 - 5.33T + 61T^{2} \)
67 \( 1 + 2.95T + 67T^{2} \)
71 \( 1 - 5.03T + 71T^{2} \)
73 \( 1 - 4.45T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 8.71T + 83T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + 5.72T + 97T^{2} \)
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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

−8.110968873672707135676872051834, −7.05414468132883204990171264533, −6.69404602905690559937072337408, −5.78121518764757116633989137746, −5.25834651713598471866123264801, −4.48246688556013077106386444843, −3.70201295754067387089369760096, −2.97191318580120063776216690433, −2.37309705355400774326344515713, −0.57156561196420703576419892950, 0.57156561196420703576419892950, 2.37309705355400774326344515713, 2.97191318580120063776216690433, 3.70201295754067387089369760096, 4.48246688556013077106386444843, 5.25834651713598471866123264801, 5.78121518764757116633989137746, 6.69404602905690559937072337408, 7.05414468132883204990171264533, 8.110968873672707135676872051834

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