Properties

Label 2-3450-1.1-c1-0-11
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 − 1.44·7-s − 8-s + 9-s + 2·11-s − 12-s + 2.89·13-s + 1.44·14-s + 16-s + 5.44·17-s − 18-s + 6.89·19-s + 1.44·21-s − 2·22-s − 23-s + 24-s − 2.89·26-s − 27-s − 1.44·28-s + 5·29-s + 2·31-s − 32-s − 2·33-s − 5.44·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.547·7-s − 0.353·8-s + 0.333·9-s + 0.603·11-s − 0.288·12-s + 0.804·13-s + 0.387·14-s + 0.250·16-s + 1.32·17-s − 0.235·18-s + 1.58·19-s + 0.316·21-s − 0.426·22-s − 0.208·23-s + 0.204·24-s − 0.568·26-s − 0.192·27-s − 0.273·28-s + 0.928·29-s + 0.359·31-s − 0.176·32-s − 0.348·33-s − 0.934·34-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

\(L(1)\) \(\approx\) \(1.174027892\)
\(L(\frac12)\) \(\approx\) \(1.174027892\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.44T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 2.89T + 13T^{2} \)
17 \( 1 - 5.44T + 17T^{2} \)
19 \( 1 - 6.89T + 19T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 8.34T + 37T^{2} \)
41 \( 1 + 4.89T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 - 3.89T + 47T^{2} \)
53 \( 1 + 0.898T + 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 4.89T + 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 2.10T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 2.55T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−8.556758322052598413455821173484, −7.939759481935455255897782339782, −6.99712430465445069179203558249, −6.52947384322782476968648320743, −5.68103501976028772010379158677, −5.01252076294990519467239837654, −3.64966530623307258319059813132, −3.15489402482554042716305666375, −1.61713147726961030963901754978, −0.78820370195956807937551751983, 0.78820370195956807937551751983, 1.61713147726961030963901754978, 3.15489402482554042716305666375, 3.64966530623307258319059813132, 5.01252076294990519467239837654, 5.68103501976028772010379158677, 6.52947384322782476968648320743, 6.99712430465445069179203558249, 7.939759481935455255897782339782, 8.556758322052598413455821173484

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