Properties

Label 2-3450-1.1-c1-0-54
Degree $2$
Conductor $3450$
Sign $-1$
Analytic cond. $27.5483$
Root an. cond. $5.24865$
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 + 3·7-s − 8-s + 9-s − 11-s − 12-s + 3·13-s − 3·14-s + 16-s − 18-s − 5·19-s − 3·21-s + 22-s − 23-s + 24-s − 3·26-s − 27-s + 3·28-s − 9·29-s − 2·31-s − 32-s + 33-s + 36-s + 5·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.832·13-s − 0.801·14-s + 1/4·16-s − 0.235·18-s − 1.14·19-s − 0.654·21-s + 0.213·22-s − 0.208·23-s + 0.204·24-s − 0.588·26-s − 0.192·27-s + 0.566·28-s − 1.67·29-s − 0.359·31-s − 0.176·32-s + 0.174·33-s + 1/6·36-s + 0.811·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(27.5483\)
Root analytic conductor: \(5.24865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
23 \( 1 + T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.250306084734283638012561628441, −7.61587244104868692281922982431, −6.80601554732646518716385931933, −6.00353478525568779160462287574, −5.30947276230842397797573574187, −4.43670963551693390439849247372, −3.51486326551566088666545372239, −2.10703971871229587508635137572, −1.43056574448160845706582086071, 0, 1.43056574448160845706582086071, 2.10703971871229587508635137572, 3.51486326551566088666545372239, 4.43670963551693390439849247372, 5.30947276230842397797573574187, 6.00353478525568779160462287574, 6.80601554732646518716385931933, 7.61587244104868692281922982431, 8.250306084734283638012561628441

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