Properties

Label 2-75e2-1.1-c1-0-60
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  + 0.141·2-s − 1.97·4-s − 0.858·7-s − 0.563·8-s + 3.67·11-s + 4.58·13-s − 0.121·14-s + 3.87·16-s + 5.30·17-s + 6.36·19-s + 0.521·22-s − 3.42·23-s + 0.649·26-s + 1.69·28-s − 3.73·29-s + 1.25·31-s + 1.67·32-s + 0.751·34-s − 7.45·37-s + 0.902·38-s + 2.53·41-s − 3.37·43-s − 7.28·44-s − 0.485·46-s + 8.49·47-s − 6.26·49-s − 9.07·52-s + ⋯
L(s)  = 1  + 0.100·2-s − 0.989·4-s − 0.324·7-s − 0.199·8-s + 1.10·11-s + 1.27·13-s − 0.0325·14-s + 0.969·16-s + 1.28·17-s + 1.46·19-s + 0.111·22-s − 0.713·23-s + 0.127·26-s + 0.321·28-s − 0.693·29-s + 0.225·31-s + 0.296·32-s + 0.128·34-s − 1.22·37-s + 0.146·38-s + 0.395·41-s − 0.515·43-s − 1.09·44-s − 0.0715·46-s + 1.23·47-s − 0.894·49-s − 1.25·52-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

\(L(1)\) \(\approx\) \(1.899262950\)
\(L(\frac12)\) \(\approx\) \(1.899262950\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.141T + 2T^{2} \)
7 \( 1 + 0.858T + 7T^{2} \)
11 \( 1 - 3.67T + 11T^{2} \)
13 \( 1 - 4.58T + 13T^{2} \)
17 \( 1 - 5.30T + 17T^{2} \)
19 \( 1 - 6.36T + 19T^{2} \)
23 \( 1 + 3.42T + 23T^{2} \)
29 \( 1 + 3.73T + 29T^{2} \)
31 \( 1 - 1.25T + 31T^{2} \)
37 \( 1 + 7.45T + 37T^{2} \)
41 \( 1 - 2.53T + 41T^{2} \)
43 \( 1 + 3.37T + 43T^{2} \)
47 \( 1 - 8.49T + 47T^{2} \)
53 \( 1 - 2.34T + 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 3.34T + 67T^{2} \)
71 \( 1 - 4.32T + 71T^{2} \)
73 \( 1 + 9.08T + 73T^{2} \)
79 \( 1 + 3.24T + 79T^{2} \)
83 \( 1 - 7.39T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−8.184941616684401270016952832355, −7.54749997491785807141597031401, −6.62963968388117209625920535925, −5.81020679726058978732641431457, −5.40147698523802868315604294169, −4.35517004591866126745666461012, −3.56718245557094863501175449574, −3.29261104145051872889558258509, −1.58269953710054108240292935980, −0.801452305009849189091906986947, 0.801452305009849189091906986947, 1.58269953710054108240292935980, 3.29261104145051872889558258509, 3.56718245557094863501175449574, 4.35517004591866126745666461012, 5.40147698523802868315604294169, 5.81020679726058978732641431457, 6.62963968388117209625920535925, 7.54749997491785807141597031401, 8.184941616684401270016952832355

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