Properties

Label 2-3450-1.1-c1-0-21
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 + 2·7-s − 8-s + 9-s + 3.50·11-s + 12-s + 2.72·13-s − 2·14-s + 16-s − 0.726·17-s − 18-s − 7.78·19-s + 2·21-s − 3.50·22-s + 23-s − 24-s − 2.72·26-s + 27-s + 2·28-s − 2·29-s + 8.28·31-s − 32-s + 3.50·33-s + 0.726·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 0.333·9-s + 1.05·11-s + 0.288·12-s + 0.756·13-s − 0.534·14-s + 0.250·16-s − 0.176·17-s − 0.235·18-s − 1.78·19-s + 0.436·21-s − 0.747·22-s + 0.208·23-s − 0.204·24-s − 0.534·26-s + 0.192·27-s + 0.377·28-s − 0.371·29-s + 1.48·31-s − 0.176·32-s + 0.610·33-s + 0.124·34-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

\(L(1)\) \(\approx\) \(2.071905280\)
\(L(\frac12)\) \(\approx\) \(2.071905280\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 3.50T + 11T^{2} \)
13 \( 1 - 2.72T + 13T^{2} \)
17 \( 1 + 0.726T + 17T^{2} \)
19 \( 1 + 7.78T + 19T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 8.28T + 31T^{2} \)
37 \( 1 - 9.50T + 37T^{2} \)
41 \( 1 - 0.726T + 41T^{2} \)
43 \( 1 - 5.50T + 43T^{2} \)
47 \( 1 - 2.72T + 47T^{2} \)
53 \( 1 + 0.231T + 53T^{2} \)
59 \( 1 - 9.29T + 59T^{2} \)
61 \( 1 - 4.05T + 61T^{2} \)
67 \( 1 - 4.05T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 - 3.71T + 73T^{2} \)
79 \( 1 + 9.73T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + 9.55T + 89T^{2} \)
97 \( 1 - 4.54T + 97T^{2} \)
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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

−8.470145758850740277189341972747, −8.211055469438346359702384527331, −7.21670277117570812067835386752, −6.49068190644895883556607253448, −5.85137686897026024416775005168, −4.46238677597533463514835924007, −4.02405890812169643812512827126, −2.78488141119260393502055790315, −1.89695051819818091337719828692, −0.981175613594656047311088880516, 0.981175613594656047311088880516, 1.89695051819818091337719828692, 2.78488141119260393502055790315, 4.02405890812169643812512827126, 4.46238677597533463514835924007, 5.85137686897026024416775005168, 6.49068190644895883556607253448, 7.21670277117570812067835386752, 8.211055469438346359702384527331, 8.470145758850740277189341972747

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