Properties

Label 2-3450-1.1-c1-0-41
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 + 4·7-s + 8-s + 9-s + 12-s − 2.47·13-s + 4·14-s + 16-s + 2.47·17-s + 18-s + 2·19-s + 4·21-s − 23-s + 24-s − 2.47·26-s + 27-s + 4·28-s − 0.472·29-s + 32-s + 2.47·34-s + 36-s + 0.472·37-s + 2·38-s − 2.47·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 0.333·9-s + 0.288·12-s − 0.685·13-s + 1.06·14-s + 0.250·16-s + 0.599·17-s + 0.235·18-s + 0.458·19-s + 0.872·21-s − 0.208·23-s + 0.204·24-s − 0.484·26-s + 0.192·27-s + 0.755·28-s − 0.0876·29-s + 0.176·32-s + 0.423·34-s + 0.166·36-s + 0.0776·37-s + 0.324·38-s − 0.395·39-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

\(L(1)\) \(\approx\) \(4.577084300\)
\(L(\frac12)\) \(\approx\) \(4.577084300\)
\(L(\frac{3}{2})\) 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 \( 1 - 4T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2.47T + 13T^{2} \)
17 \( 1 - 2.47T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
29 \( 1 + 0.472T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 0.472T + 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 4.94T + 47T^{2} \)
53 \( 1 + 8.94T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 0.472T + 61T^{2} \)
67 \( 1 + 4.94T + 67T^{2} \)
71 \( 1 + 7.52T + 71T^{2} \)
73 \( 1 - 4.94T + 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 - 1.52T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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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

−8.424193300893048907259665118448, −7.59777646235107738865208239945, −7.47318900460785230539590292658, −6.22210519152707673983040126127, −5.37714618032900714608010628465, −4.71980287745109039196904565568, −4.05934126312131973655591662643, −3.01309112498491954557173784436, −2.16627085825204070327855970572, −1.24362686956506664914624916094, 1.24362686956506664914624916094, 2.16627085825204070327855970572, 3.01309112498491954557173784436, 4.05934126312131973655591662643, 4.71980287745109039196904565568, 5.37714618032900714608010628465, 6.22210519152707673983040126127, 7.47318900460785230539590292658, 7.59777646235107738865208239945, 8.424193300893048907259665118448

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