Properties

Label 2-75e2-1.1-c1-0-97
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  − 1.52·2-s + 0.318·4-s + 0.990·7-s + 2.56·8-s − 5.97·11-s − 4.02·13-s − 1.50·14-s − 4.53·16-s + 0.476·17-s + 2.82·19-s + 9.09·22-s − 1.74·23-s + 6.13·26-s + 0.315·28-s + 1.41·29-s + 8.76·31-s + 1.78·32-s − 0.725·34-s + 8.06·37-s − 4.29·38-s + 5.50·41-s + 6.82·43-s − 1.90·44-s + 2.65·46-s − 9.62·47-s − 6.01·49-s − 1.28·52-s + ⋯
L(s)  = 1  − 1.07·2-s + 0.159·4-s + 0.374·7-s + 0.905·8-s − 1.80·11-s − 1.11·13-s − 0.403·14-s − 1.13·16-s + 0.115·17-s + 0.647·19-s + 1.93·22-s − 0.364·23-s + 1.20·26-s + 0.0596·28-s + 0.263·29-s + 1.57·31-s + 0.315·32-s − 0.124·34-s + 1.32·37-s − 0.697·38-s + 0.859·41-s + 1.04·43-s − 0.286·44-s + 0.392·46-s − 1.40·47-s − 0.859·49-s − 0.177·52-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: \(1\)
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 \( 1 + 1.52T + 2T^{2} \)
7 \( 1 - 0.990T + 7T^{2} \)
11 \( 1 + 5.97T + 11T^{2} \)
13 \( 1 + 4.02T + 13T^{2} \)
17 \( 1 - 0.476T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 1.74T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 8.76T + 31T^{2} \)
37 \( 1 - 8.06T + 37T^{2} \)
41 \( 1 - 5.50T + 41T^{2} \)
43 \( 1 - 6.82T + 43T^{2} \)
47 \( 1 + 9.62T + 47T^{2} \)
53 \( 1 - 6.57T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 - 5.43T + 71T^{2} \)
73 \( 1 - 4.08T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + 2.29T + 83T^{2} \)
89 \( 1 + 6.26T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−7.954274822006941075419440180092, −7.50311098597086317454427770200, −6.57979758990905918983530937206, −5.50153414200802299458402499753, −4.89812748539917079706314553346, −4.29753178827293006434980242660, −2.86262142231049834533115228714, −2.30136488946697099526484441480, −1.04985205671398068911079983900, 0, 1.04985205671398068911079983900, 2.30136488946697099526484441480, 2.86262142231049834533115228714, 4.29753178827293006434980242660, 4.89812748539917079706314553346, 5.50153414200802299458402499753, 6.57979758990905918983530937206, 7.50311098597086317454427770200, 7.954274822006941075419440180092

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