Properties

Label 2-3450-1.1-c1-0-18
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 − 2.77·11-s − 12-s − 4.77·13-s + 3·14-s + 16-s + 7.77·17-s + 18-s − 0.772·19-s − 3·21-s − 2.77·22-s − 23-s − 24-s − 4.77·26-s − 27-s + 3·28-s + 3·29-s + 7.54·31-s + 32-s + 2.77·33-s + 7.77·34-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 − 0.835·11-s − 0.288·12-s − 1.32·13-s + 0.801·14-s + 0.250·16-s + 1.88·17-s + 0.235·18-s − 0.177·19-s − 0.654·21-s − 0.590·22-s − 0.208·23-s − 0.204·24-s − 0.935·26-s − 0.192·27-s + 0.566·28-s + 0.557·29-s + 1.35·31-s + 0.176·32-s + 0.482·33-s + 1.33·34-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

\(L(1)\) \(\approx\) \(2.722373503\)
\(L(\frac12)\) \(\approx\) \(2.722373503\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.77T + 11T^{2} \)
13 \( 1 + 4.77T + 13T^{2} \)
17 \( 1 - 7.77T + 17T^{2} \)
19 \( 1 + 0.772T + 19T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 7.54T + 31T^{2} \)
37 \( 1 - 1.77T + 37T^{2} \)
41 \( 1 - 2.77T + 41T^{2} \)
43 \( 1 - 0.772T + 43T^{2} \)
47 \( 1 - 0.227T + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 7.54T + 61T^{2} \)
67 \( 1 + 7.54T + 67T^{2} \)
71 \( 1 - 3.77T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 - 8.22T + 89T^{2} \)
97 \( 1 + 0.455T + 97T^{2} \)
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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

−8.116052975104004691139896233720, −7.896864887521340210683875929103, −7.10164789986550721222323650391, −6.16405788331411442844935596233, −5.25818603579471363273071133104, −5.02273631098538560763820473238, −4.18442320884800096427412227888, −3.00709943307207086653481044016, −2.14584077756646401615564170703, −0.930496308390897059512022215609, 0.930496308390897059512022215609, 2.14584077756646401615564170703, 3.00709943307207086653481044016, 4.18442320884800096427412227888, 5.02273631098538560763820473238, 5.25818603579471363273071133104, 6.16405788331411442844935596233, 7.10164789986550721222323650391, 7.896864887521340210683875929103, 8.116052975104004691139896233720

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