Properties

Label 2-3675-1.1-c1-0-94
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  − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s − 12-s − 6·13-s − 16-s + 2·17-s − 18-s + 8·19-s − 8·23-s + 3·24-s + 6·26-s + 27-s − 2·29-s − 4·31-s − 5·32-s − 2·34-s − 36-s + 2·37-s − 8·38-s − 6·39-s + 6·41-s − 4·43-s + 8·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.288·12-s − 1.66·13-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.83·19-s − 1.66·23-s + 0.612·24-s + 1.17·26-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.883·32-s − 0.342·34-s − 1/6·36-s + 0.328·37-s − 1.29·38-s − 0.960·39-s + 0.937·41-s − 0.609·43-s + 1.17·46-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 \( 1 + T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.006648689764812283420657294904, −7.66078399107701284311775872096, −7.16566744133333875383333517560, −5.82680289170118734614021714302, −5.07896009712412898647893372722, −4.30076158847618591361657861041, −3.40922618253584977154524124894, −2.38993500725665786110797187633, −1.34577586096679437279208168488, 0, 1.34577586096679437279208168488, 2.38993500725665786110797187633, 3.40922618253584977154524124894, 4.30076158847618591361657861041, 5.07896009712412898647893372722, 5.82680289170118734614021714302, 7.16566744133333875383333517560, 7.66078399107701284311775872096, 8.006648689764812283420657294904

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