Properties

Label 2-3450-1.1-c1-0-63
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 2.96·7-s − 8-s + 9-s − 3.35·11-s + 12-s − 4.96·13-s − 2.96·14-s + 16-s + 1.35·17-s − 18-s − 4.96·19-s + 2.96·21-s + 3.35·22-s − 23-s − 24-s + 4.96·26-s + 27-s + 2.96·28-s + 7.73·29-s − 4·31-s − 32-s − 3.35·33-s − 1.35·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.11·7-s − 0.353·8-s + 0.333·9-s − 1.01·11-s + 0.288·12-s − 1.37·13-s − 0.791·14-s + 0.250·16-s + 0.327·17-s − 0.235·18-s − 1.13·19-s + 0.646·21-s + 0.714·22-s − 0.208·23-s − 0.204·24-s + 0.973·26-s + 0.192·27-s + 0.559·28-s + 1.43·29-s − 0.718·31-s − 0.176·32-s − 0.583·33-s − 0.231·34-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: \(1\)
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 \( 1 - 2.96T + 7T^{2} \)
11 \( 1 + 3.35T + 11T^{2} \)
13 \( 1 + 4.96T + 13T^{2} \)
17 \( 1 - 1.35T + 17T^{2} \)
19 \( 1 + 4.96T + 19T^{2} \)
29 \( 1 - 7.73T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 7.61T + 37T^{2} \)
41 \( 1 - 4.70T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 - 3.22T + 47T^{2} \)
53 \( 1 + 6.96T + 53T^{2} \)
59 \( 1 - 1.22T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 7.61T + 67T^{2} \)
71 \( 1 + 2.18T + 71T^{2} \)
73 \( 1 + 9.92T + 73T^{2} \)
79 \( 1 + 4.12T + 79T^{2} \)
83 \( 1 + 6.38T + 83T^{2} \)
89 \( 1 - 9.92T + 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−8.283013254583847062558932925473, −7.62009069527465232948193517951, −7.13190996567842549811751982294, −6.06324685912821118957369484650, −4.99962534891339095194888744620, −4.54692251220821180241167425042, −3.17972308835723140094762879940, −2.34525011462089560304973847153, −1.61446291174717628369372889735, 0, 1.61446291174717628369372889735, 2.34525011462089560304973847153, 3.17972308835723140094762879940, 4.54692251220821180241167425042, 4.99962534891339095194888744620, 6.06324685912821118957369484650, 7.13190996567842549811751982294, 7.62009069527465232948193517951, 8.283013254583847062558932925473

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