Properties

Label 2-3450-1.1-c1-0-40
Degree $2$
Conductor $3450$
Sign $1$
Analytic cond. $27.5483$
Root an. cond. $5.24865$
Motivic weight $1$
Arithmetic yes
Rational no
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 + 3·7-s + 8-s + 9-s − 1.27·11-s + 12-s + 3.27·13-s + 3·14-s + 16-s − 2.27·17-s + 18-s + 3.27·19-s + 3·21-s − 1.27·22-s − 23-s + 24-s + 3.27·26-s + 27-s + 3·28-s − 9.54·29-s + 6·31-s + 32-s − 1.27·33-s − 2.27·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 0.333·9-s − 0.384·11-s + 0.288·12-s + 0.908·13-s + 0.801·14-s + 0.250·16-s − 0.551·17-s + 0.235·18-s + 0.751·19-s + 0.654·21-s − 0.271·22-s − 0.208·23-s + 0.204·24-s + 0.642·26-s + 0.192·27-s + 0.566·28-s − 1.77·29-s + 1.07·31-s + 0.176·32-s − 0.221·33-s − 0.390·34-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: no
Arithmetic: yes
Character: $\chi_{3450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.474716513\)
\(L(\frac12)\) \(\approx\) \(4.474716513\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T + 7T^{2} \)
11 \( 1 + 1.27T + 11T^{2} \)
13 \( 1 - 3.27T + 13T^{2} \)
17 \( 1 + 2.27T + 17T^{2} \)
19 \( 1 - 3.27T + 19T^{2} \)
29 \( 1 + 9.54T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 + 2.72T + 41T^{2} \)
43 \( 1 - 7.27T + 43T^{2} \)
47 \( 1 - 8.27T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 0.549T + 61T^{2} \)
67 \( 1 + 4.54T + 67T^{2} \)
71 \( 1 + 8.82T + 71T^{2} \)
73 \( 1 + 15.5T + 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + 2.45T + 83T^{2} \)
89 \( 1 - 8.27T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−8.482349470207487994089384585087, −7.74356215802293645368635834715, −7.30915701166229494922937978900, −6.15613081488162268823057584799, −5.55667873776478426197912082623, −4.59007998762518450904543904154, −4.06174804262391857785637188679, −3.05167584540746076079639218932, −2.17068861923519461448644686088, −1.22073454539954074236306540924, 1.22073454539954074236306540924, 2.17068861923519461448644686088, 3.05167584540746076079639218932, 4.06174804262391857785637188679, 4.59007998762518450904543904154, 5.55667873776478426197912082623, 6.15613081488162268823057584799, 7.30915701166229494922937978900, 7.74356215802293645368635834715, 8.482349470207487994089384585087

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