Properties

Label 2-3450-1.1-c1-0-27
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 − 3.44·7-s + 8-s + 9-s + 2·11-s + 12-s + 6.89·13-s − 3.44·14-s + 16-s − 0.550·17-s + 18-s − 2.89·19-s − 3.44·21-s + 2·22-s + 23-s + 24-s + 6.89·26-s + 27-s − 3.44·28-s + 5·29-s + 2·31-s + 32-s + 2·33-s − 0.550·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.30·7-s + 0.353·8-s + 0.333·9-s + 0.603·11-s + 0.288·12-s + 1.91·13-s − 0.921·14-s + 0.250·16-s − 0.133·17-s + 0.235·18-s − 0.665·19-s − 0.752·21-s + 0.426·22-s + 0.208·23-s + 0.204·24-s + 1.35·26-s + 0.192·27-s − 0.651·28-s + 0.928·29-s + 0.359·31-s + 0.176·32-s + 0.348·33-s − 0.0944·34-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

\(L(1)\) \(\approx\) \(3.664761143\)
\(L(\frac12)\) \(\approx\) \(3.664761143\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.44T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 6.89T + 13T^{2} \)
17 \( 1 + 0.550T + 17T^{2} \)
19 \( 1 + 2.89T + 19T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 6.34T + 37T^{2} \)
41 \( 1 - 4.89T + 41T^{2} \)
43 \( 1 - 1.10T + 43T^{2} \)
47 \( 1 - 5.89T + 47T^{2} \)
53 \( 1 + 8.89T + 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 - 4.89T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + 8.79T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 7.44T + 83T^{2} \)
89 \( 1 - 4.34T + 89T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−8.735757112643619858699934952396, −7.85158362548015781399955165410, −6.73517043472866297590925331081, −6.45502175381986450532353212218, −5.75152047587205172334958862606, −4.53657530114506929698115319560, −3.70871859967191048066391636922, −3.32078490551390656244289907353, −2.27940906816759411741271149710, −1.04075523359927177757821732563, 1.04075523359927177757821732563, 2.27940906816759411741271149710, 3.32078490551390656244289907353, 3.70871859967191048066391636922, 4.53657530114506929698115319560, 5.75152047587205172334958862606, 6.45502175381986450532353212218, 6.73517043472866297590925331081, 7.85158362548015781399955165410, 8.735757112643619858699934952396

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