Properties

Label 2-3450-1.1-c1-0-56
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 − 8-s + 9-s + 12-s − 6·13-s + 16-s − 2·17-s − 18-s + 23-s − 24-s + 6·26-s + 27-s + 6·29-s + 8·31-s − 32-s + 2·34-s + 36-s − 10·37-s − 6·39-s − 6·41-s + 8·43-s − 46-s − 8·47-s + 48-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.208·23-s − 0.204·24-s + 1.17·26-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 1/6·36-s − 1.64·37-s − 0.960·39-s − 0.937·41-s + 1.21·43-s − 0.147·46-s − 1.16·47-s + 0.144·48-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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.363091242101441457816037482401, −7.51934293395641500406980628219, −6.98901007484807183874746617029, −6.23264379881818797359793277018, −5.04138360934187634925916833280, −4.43309662625513659949140379055, −3.12737656626193666211958659956, −2.52625422200265821165388851985, −1.49406226657518668683883312451, 0, 1.49406226657518668683883312451, 2.52625422200265821165388851985, 3.12737656626193666211958659956, 4.43309662625513659949140379055, 5.04138360934187634925916833280, 6.23264379881818797359793277018, 6.98901007484807183874746617029, 7.51934293395641500406980628219, 8.363091242101441457816037482401

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