Properties

Label 2-3675-1.1-c1-0-61
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 − 6·11-s + 12-s + 2·13-s − 16-s − 4·17-s − 18-s + 6·19-s + 6·22-s − 3·24-s − 2·26-s − 27-s − 2·29-s + 10·31-s − 5·32-s + 6·33-s + 4·34-s − 36-s − 4·37-s − 6·38-s − 2·39-s − 2·41-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 − 1.80·11-s + 0.288·12-s + 0.554·13-s − 1/4·16-s − 0.970·17-s − 0.235·18-s + 1.37·19-s + 1.27·22-s − 0.612·24-s − 0.392·26-s − 0.192·27-s − 0.371·29-s + 1.79·31-s − 0.883·32-s + 1.04·33-s + 0.685·34-s − 1/6·36-s − 0.657·37-s − 0.973·38-s − 0.320·39-s − 0.312·41-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.256055578461575445003684081516, −7.55446969326170854719470118234, −6.87365214021075367013688836973, −5.80251515017269393523381865598, −5.11647068828854224957302425430, −4.56813311176270756452064483969, −3.45367463617503835925378630300, −2.33462003232101750035985685627, −1.06504307972466057410251062188, 0, 1.06504307972466057410251062188, 2.33462003232101750035985685627, 3.45367463617503835925378630300, 4.56813311176270756452064483969, 5.11647068828854224957302425430, 5.80251515017269393523381865598, 6.87365214021075367013688836973, 7.55446969326170854719470118234, 8.256055578461575445003684081516

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