Properties

Label 2-3675-1.1-c1-0-103
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

−8.135245437116732921186666216165, −7.75110803385765164493233092171, −6.75870879468165706986788693687, −5.71712970663256425424326189478, −5.17862009613309092684526088674, −4.02346664539105473677147683060, −3.74662313532088859915068806107, −2.55130249126529936360206208492, −1.44711402334891860578104024553, 0, 1.44711402334891860578104024553, 2.55130249126529936360206208492, 3.74662313532088859915068806107, 4.02346664539105473677147683060, 5.17862009613309092684526088674, 5.71712970663256425424326189478, 6.75870879468165706986788693687, 7.75110803385765164493233092171, 8.135245437116732921186666216165

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