Properties

Label 2-3234-1.1-c1-0-15
Degree $2$
Conductor $3234$
Sign $1$
Analytic cond. $25.8236$
Root an. cond. $5.08169$
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 − 8-s + 9-s − 11-s − 12-s + 4·13-s + 16-s + 6·17-s − 18-s + 4·19-s + 22-s + 6·23-s + 24-s − 5·25-s − 4·26-s − 27-s + 6·29-s − 8·31-s − 32-s + 33-s − 6·34-s + 36-s − 10·37-s − 4·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.213·22-s + 1.25·23-s + 0.204·24-s − 25-s − 0.784·26-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.174·33-s − 1.02·34-s + 1/6·36-s − 1.64·37-s − 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(25.8236\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3234,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.181910159\)
\(L(\frac12)\) \(\approx\) \(1.181910159\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good5 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.731622022691464682737122465780, −7.82645545698547241323210264911, −7.31156938226239113875453585273, −6.46285019703242959678902869410, −5.62122186770240684074690473279, −5.14110181688607884712664879052, −3.77174402308413079522587358569, −3.08375157042371119503272456681, −1.67169823291829433403188663441, −0.792903116476162912230355051834, 0.792903116476162912230355051834, 1.67169823291829433403188663441, 3.08375157042371119503272456681, 3.77174402308413079522587358569, 5.14110181688607884712664879052, 5.62122186770240684074690473279, 6.46285019703242959678902869410, 7.31156938226239113875453585273, 7.82645545698547241323210264911, 8.731622022691464682737122465780

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