Properties

Label 2-3420-1.1-c1-0-19
Degree $2$
Conductor $3420$
Sign $-1$
Analytic cond. $27.3088$
Root an. cond. $5.22578$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·7-s − 2·11-s + 6·13-s + 2·17-s + 19-s − 6·23-s + 25-s − 8·29-s − 8·31-s − 4·35-s + 10·37-s + 4·41-s + 4·43-s − 6·47-s + 9·49-s − 12·53-s − 2·55-s + 8·59-s + 2·61-s + 6·65-s + 4·67-s − 12·71-s − 10·73-s + 8·77-s − 8·79-s − 2·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s − 0.603·11-s + 1.66·13-s + 0.485·17-s + 0.229·19-s − 1.25·23-s + 1/5·25-s − 1.48·29-s − 1.43·31-s − 0.676·35-s + 1.64·37-s + 0.624·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s − 1.64·53-s − 0.269·55-s + 1.04·59-s + 0.256·61-s + 0.744·65-s + 0.488·67-s − 1.42·71-s − 1.17·73-s + 0.911·77-s − 0.900·79-s − 0.219·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
19 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−8.195154278027595287101589658656, −7.51763594068408576761934425881, −6.57264711987684388402294646931, −5.89240349375805042329343407251, −5.58682683235758880385578107278, −4.10107534324134400504475256340, −3.50631479190343971424499020931, −2.65176951800295550388157692815, −1.45537436864446162809269751533, 0, 1.45537436864446162809269751533, 2.65176951800295550388157692815, 3.50631479190343971424499020931, 4.10107534324134400504475256340, 5.58682683235758880385578107278, 5.89240349375805042329343407251, 6.57264711987684388402294646931, 7.51763594068408576761934425881, 8.195154278027595287101589658656

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