Properties

Label 2-6525-1.1-c1-0-60
Degree $2$
Conductor $6525$
Sign $1$
Analytic cond. $52.1023$
Root an. cond. $7.21819$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 4·7-s + 3·8-s − 6·13-s − 4·14-s − 16-s + 2·17-s + 8·19-s − 4·23-s + 6·26-s − 4·28-s − 29-s + 4·31-s − 5·32-s − 2·34-s − 6·37-s − 8·38-s − 2·41-s + 4·43-s + 4·46-s + 9·49-s + 6·52-s + 6·53-s + 12·56-s + 58-s + 12·59-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.51·7-s + 1.06·8-s − 1.66·13-s − 1.06·14-s − 1/4·16-s + 0.485·17-s + 1.83·19-s − 0.834·23-s + 1.17·26-s − 0.755·28-s − 0.185·29-s + 0.718·31-s − 0.883·32-s − 0.342·34-s − 0.986·37-s − 1.29·38-s − 0.312·41-s + 0.609·43-s + 0.589·46-s + 9/7·49-s + 0.832·52-s + 0.824·53-s + 1.60·56-s + 0.131·58-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6525\)    =    \(3^{2} \cdot 5^{2} \cdot 29\)
Sign: $1$
Analytic conductor: \(52.1023\)
Root analytic conductor: \(7.21819\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6525,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.331009955\)
\(L(\frac12)\) \(\approx\) \(1.331009955\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.045362182214987600521266034391, −7.47213944209812981166880659148, −7.06774260128796458557570773193, −5.54862256182882684732427206128, −5.17138216573938377921881410546, −4.56798076262690059135394375858, −3.71170366344554859751177680982, −2.49716365422468470811732516052, −1.61676034999860228513231780102, −0.70750026455756233727087866463, 0.70750026455756233727087866463, 1.61676034999860228513231780102, 2.49716365422468470811732516052, 3.71170366344554859751177680982, 4.56798076262690059135394375858, 5.17138216573938377921881410546, 5.54862256182882684732427206128, 7.06774260128796458557570773193, 7.47213944209812981166880659148, 8.045362182214987600521266034391

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