Properties

Label 2-3450-1.1-c1-0-43
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 + 7-s − 8-s + 9-s − 5·11-s − 12-s + 3·13-s − 14-s + 16-s + 2·17-s − 18-s − 3·19-s − 21-s + 5·22-s + 23-s + 24-s − 3·26-s − 27-s + 28-s − 29-s − 32-s + 5·33-s − 2·34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.688·19-s − 0.218·21-s + 1.06·22-s + 0.208·23-s + 0.204·24-s − 0.588·26-s − 0.192·27-s + 0.188·28-s − 0.185·29-s − 0.176·32-s + 0.870·33-s − 0.342·34-s + 1/6·36-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 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−8.102141859594489834185502774130, −7.71366517870895122381147621219, −6.77062538829263091086808872333, −6.00048317520766404009010192364, −5.32459672559986710113356864090, −4.50267108293831127901609980272, −3.34939182019830834851839432972, −2.33985608070019533541704094676, −1.26727594197553854827460911547, 0, 1.26727594197553854827460911547, 2.33985608070019533541704094676, 3.34939182019830834851839432972, 4.50267108293831127901609980272, 5.32459672559986710113356864090, 6.00048317520766404009010192364, 6.77062538829263091086808872333, 7.71366517870895122381147621219, 8.102141859594489834185502774130

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