Properties

Label 2-3450-1.1-c1-0-28
Degree $2$
Conductor $3450$
Sign $1$
Analytic cond. $27.5483$
Root an. cond. $5.24865$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 7-s + 8-s + 9-s + 12-s + 2·13-s − 14-s + 16-s + 3·17-s + 18-s + 2·19-s − 21-s − 23-s + 24-s + 2·26-s + 27-s − 28-s + 9·29-s − 10·31-s + 32-s + 3·34-s + 36-s + 11·37-s + 2·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.67·29-s − 1.79·31-s + 0.176·32-s + 0.514·34-s + 1/6·36-s + 1.80·37-s + 0.324·38-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: yes
Arithmetic: yes
Character: $\chi_{3450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.886406483\)
\(L(\frac12)\) \(\approx\) \(3.886406483\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.349300711835563670740425205988, −7.937156705554867691859168622097, −6.99535482217755614986865443256, −6.34516048850235012007737676278, −5.54429847923358013611830820295, −4.70586181675141721672120670621, −3.76832686949922865967787924483, −3.19556541482137519648986671948, −2.27712057566840263929378719324, −1.08409241778139482879615046752, 1.08409241778139482879615046752, 2.27712057566840263929378719324, 3.19556541482137519648986671948, 3.76832686949922865967787924483, 4.70586181675141721672120670621, 5.54429847923358013611830820295, 6.34516048850235012007737676278, 6.99535482217755614986865443256, 7.937156705554867691859168622097, 8.349300711835563670740425205988

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