Properties

Label 2-75e2-1.1-c1-0-129
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.618·2-s − 1.61·4-s − 2·7-s − 2.23·8-s + 3·11-s − 13-s − 1.23·14-s + 1.85·16-s − 0.236·17-s + 6.70·19-s + 1.85·22-s − 7.61·23-s − 0.618·26-s + 3.23·28-s + 1.38·29-s − 4.70·31-s + 5.61·32-s − 0.145·34-s − 2·37-s + 4.14·38-s + 11.6·41-s − 9.61·43-s − 4.85·44-s − 4.70·46-s + 9.23·47-s − 3·49-s + 1.61·52-s + ⋯
L(s)  = 1  + 0.437·2-s − 0.809·4-s − 0.755·7-s − 0.790·8-s + 0.904·11-s − 0.277·13-s − 0.330·14-s + 0.463·16-s − 0.0572·17-s + 1.53·19-s + 0.395·22-s − 1.58·23-s − 0.121·26-s + 0.611·28-s + 0.256·29-s − 0.845·31-s + 0.993·32-s − 0.0250·34-s − 0.328·37-s + 0.672·38-s + 1.81·41-s − 1.46·43-s − 0.731·44-s − 0.694·46-s + 1.34·47-s − 0.428·49-s + 0.224·52-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: $\chi_{5625} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5625,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.618T + 2T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + 0.236T + 17T^{2} \)
19 \( 1 - 6.70T + 19T^{2} \)
23 \( 1 + 7.61T + 23T^{2} \)
29 \( 1 - 1.38T + 29T^{2} \)
31 \( 1 + 4.70T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 + 9.61T + 43T^{2} \)
47 \( 1 - 9.23T + 47T^{2} \)
53 \( 1 + 6.76T + 53T^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 + 4.70T + 61T^{2} \)
67 \( 1 - 9.18T + 67T^{2} \)
71 \( 1 - 1.09T + 71T^{2} \)
73 \( 1 - 2.29T + 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 + 2.85T + 97T^{2} \)
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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

−7.75574510743999089354799921660, −7.02153086636121786566493303414, −6.16140025949265467011638416969, −5.64670458012498577193518659701, −4.84624068738795639397808224097, −3.93205671171735489237269264874, −3.54728059228494985462628079567, −2.56768649451905884485716367257, −1.22155713237536775014526789573, 0, 1.22155713237536775014526789573, 2.56768649451905884485716367257, 3.54728059228494985462628079567, 3.93205671171735489237269264874, 4.84624068738795639397808224097, 5.64670458012498577193518659701, 6.16140025949265467011638416969, 7.02153086636121786566493303414, 7.75574510743999089354799921660

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