Properties

Label 2-3450-1.1-c1-0-62
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 − 0.622·7-s − 8-s + 9-s + 4.42·11-s + 12-s − 1.37·13-s + 0.622·14-s + 16-s − 6.42·17-s − 18-s − 1.37·19-s − 0.622·21-s − 4.42·22-s − 23-s − 24-s + 1.37·26-s + 27-s − 0.622·28-s − 4.23·29-s − 4·31-s − 32-s + 4.42·33-s + 6.42·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.235·7-s − 0.353·8-s + 0.333·9-s + 1.33·11-s + 0.288·12-s − 0.382·13-s + 0.166·14-s + 0.250·16-s − 1.55·17-s − 0.235·18-s − 0.316·19-s − 0.135·21-s − 0.944·22-s − 0.208·23-s − 0.204·24-s + 0.270·26-s + 0.192·27-s − 0.117·28-s − 0.786·29-s − 0.718·31-s − 0.176·32-s + 0.770·33-s + 1.10·34-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 \( 1 + 0.622T + 7T^{2} \)
11 \( 1 - 4.42T + 11T^{2} \)
13 \( 1 + 1.37T + 13T^{2} \)
17 \( 1 + 6.42T + 17T^{2} \)
19 \( 1 + 1.37T + 19T^{2} \)
29 \( 1 + 4.23T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 1.05T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 3.37T + 53T^{2} \)
59 \( 1 - 9.61T + 59T^{2} \)
61 \( 1 - 8.66T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 + 6.99T + 71T^{2} \)
73 \( 1 + 2.75T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + 2.19T + 83T^{2} \)
89 \( 1 - 2.75T + 89T^{2} \)
97 \( 1 - 2.13T + 97T^{2} \)
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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

−8.559359405928602889981414889651, −7.43676034073385431128546156090, −6.87638009373953689059645306842, −6.32297653828020260530502818724, −5.18873584588930821924570432981, −4.11315054453020328279282118304, −3.47077496089904345466788332906, −2.28196805559722431665587707675, −1.58437240387044442982921760080, 0, 1.58437240387044442982921760080, 2.28196805559722431665587707675, 3.47077496089904345466788332906, 4.11315054453020328279282118304, 5.18873584588930821924570432981, 6.32297653828020260530502818724, 6.87638009373953689059645306842, 7.43676034073385431128546156090, 8.559359405928602889981414889651

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