Properties

Label 2-3450-1.1-c1-0-49
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 5·7-s + 8-s + 9-s − 3·11-s + 12-s + 5·13-s + 5·14-s + 16-s + 6·17-s + 18-s − 19-s + 5·21-s − 3·22-s + 23-s + 24-s + 5·26-s + 27-s + 5·28-s − 5·29-s − 8·31-s + 32-s − 3·33-s + 6·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.88·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s + 1.38·13-s + 1.33·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.229·19-s + 1.09·21-s − 0.639·22-s + 0.208·23-s + 0.204·24-s + 0.980·26-s + 0.192·27-s + 0.944·28-s − 0.928·29-s − 1.43·31-s + 0.176·32-s − 0.522·33-s + 1.02·34-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: \(0\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.887565544\)
\(L(\frac12)\) \(\approx\) \(4.887565544\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−8.372009791103050200928030615094, −7.81289319692928862111948230881, −7.39365622616854088162652204664, −6.14761192235402227626233687219, −5.35167504517160095125248916943, −4.87584199196971761824060632897, −3.86267712703084611541901458826, −3.20577103932143191620076692294, −1.97934929128960217007044830278, −1.36009932110755763861706539176, 1.36009932110755763861706539176, 1.97934929128960217007044830278, 3.20577103932143191620076692294, 3.86267712703084611541901458826, 4.87584199196971761824060632897, 5.35167504517160095125248916943, 6.14761192235402227626233687219, 7.39365622616854088162652204664, 7.81289319692928862111948230881, 8.372009791103050200928030615094

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