Properties

Label 2-3675-1.1-c1-0-77
Degree $2$
Conductor $3675$
Sign $-1$
Analytic cond. $29.3450$
Root an. cond. $5.41710$
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  − 0.193·2-s − 3-s − 1.96·4-s + 0.193·6-s + 0.768·8-s + 9-s + 2·11-s + 1.96·12-s + 1.35·13-s + 3.77·16-s − 3.35·17-s − 0.193·18-s − 5.35·19-s − 0.387·22-s − 4.96·23-s − 0.768·24-s − 0.261·26-s − 27-s + 7.92·29-s − 4.57·31-s − 2.26·32-s − 2·33-s + 0.649·34-s − 1.96·36-s + 0.775·37-s + 1.03·38-s − 1.35·39-s + ⋯
L(s)  = 1  − 0.137·2-s − 0.577·3-s − 0.981·4-s + 0.0791·6-s + 0.271·8-s + 0.333·9-s + 0.603·11-s + 0.566·12-s + 0.374·13-s + 0.943·16-s − 0.812·17-s − 0.0457·18-s − 1.22·19-s − 0.0826·22-s − 1.03·23-s − 0.156·24-s − 0.0513·26-s − 0.192·27-s + 1.47·29-s − 0.821·31-s − 0.401·32-s − 0.348·33-s + 0.111·34-s − 0.327·36-s + 0.127·37-s + 0.168·38-s − 0.216·39-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: no
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 \( 1 + 0.193T + 2T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 1.35T + 13T^{2} \)
17 \( 1 + 3.35T + 17T^{2} \)
19 \( 1 + 5.35T + 19T^{2} \)
23 \( 1 + 4.96T + 23T^{2} \)
29 \( 1 - 7.92T + 29T^{2} \)
31 \( 1 + 4.57T + 31T^{2} \)
37 \( 1 - 0.775T + 37T^{2} \)
41 \( 1 + 3.73T + 41T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 - 9.92T + 47T^{2} \)
53 \( 1 + 8.57T + 53T^{2} \)
59 \( 1 - 8.62T + 59T^{2} \)
61 \( 1 - 8.70T + 61T^{2} \)
67 \( 1 + 9.92T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 9.35T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 3.22T + 83T^{2} \)
89 \( 1 + 1.03T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−8.345802947219802098136539132955, −7.46757460100901602501198410287, −6.50584057956923215638309814261, −5.99388072525571371806316475364, −5.07799165705330983961719586369, −4.21910684655347965701270558245, −3.88439785825654965994471494230, −2.39346676218986079088494122679, −1.17649483622796969392420270617, 0, 1.17649483622796969392420270617, 2.39346676218986079088494122679, 3.88439785825654965994471494230, 4.21910684655347965701270558245, 5.07799165705330983961719586369, 5.99388072525571371806316475364, 6.50584057956923215638309814261, 7.46757460100901602501198410287, 8.345802947219802098136539132955

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