Properties

Label 2-3675-1.1-c1-0-76
Degree $2$
Conductor $3675$
Sign $-1$
Analytic cond. $29.3450$
Root an. cond. $5.41710$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s + 9-s + 2·12-s + 13-s + 4·16-s − 6·17-s + 5·19-s − 6·23-s − 27-s − 6·29-s + 5·31-s − 2·36-s + 7·37-s − 39-s + 12·41-s + 43-s − 6·47-s − 4·48-s + 6·51-s − 2·52-s − 5·57-s − 6·59-s + 2·61-s − 8·64-s + 7·67-s + 12·68-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 1/3·9-s + 0.577·12-s + 0.277·13-s + 16-s − 1.45·17-s + 1.14·19-s − 1.25·23-s − 0.192·27-s − 1.11·29-s + 0.898·31-s − 1/3·36-s + 1.15·37-s − 0.160·39-s + 1.87·41-s + 0.152·43-s − 0.875·47-s − 0.577·48-s + 0.840·51-s − 0.277·52-s − 0.662·57-s − 0.781·59-s + 0.256·61-s − 64-s + 0.855·67-s + 1.45·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(29.3450\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3675} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3675,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.075858396034009908883587475553, −7.60126342338083394994512253470, −6.49076701475325398744078368158, −5.89061496774930711493236666403, −5.10329659674794759935996897576, −4.33616884277961121625575105132, −3.76171271426442357362606688289, −2.49160116958784079825199715227, −1.17137039860218412081652255199, 0, 1.17137039860218412081652255199, 2.49160116958784079825199715227, 3.76171271426442357362606688289, 4.33616884277961121625575105132, 5.10329659674794759935996897576, 5.89061496774930711493236666403, 6.49076701475325398744078368158, 7.60126342338083394994512253470, 8.075858396034009908883587475553

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