Properties

Label 2-334620-1.1-c1-0-26
Degree $2$
Conductor $334620$
Sign $-1$
Analytic cond. $2671.95$
Root an. cond. $51.6909$
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 − 11-s − 3·17-s − 4·19-s − 6·23-s + 25-s − 6·29-s − 7·31-s + 8·37-s + 8·41-s + 43-s + 2·47-s − 7·49-s + 12·53-s + 55-s + 3·59-s + 2·67-s − 8·71-s − 3·73-s − 8·79-s + 9·83-s + 3·85-s − 11·89-s + 4·95-s + 8·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s − 0.727·17-s − 0.917·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 1.25·31-s + 1.31·37-s + 1.24·41-s + 0.152·43-s + 0.291·47-s − 49-s + 1.64·53-s + 0.134·55-s + 0.390·59-s + 0.244·67-s − 0.949·71-s − 0.351·73-s − 0.900·79-s + 0.987·83-s + 0.325·85-s − 1.16·89-s + 0.410·95-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(334620\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(2671.95\)
Root analytic conductor: \(51.6909\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 334620,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 \)
good7 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−12.88296072866546, −12.47517697162411, −11.80637252971138, −11.44502604265131, −10.99678754229040, −10.68615837484667, −10.07536175726445, −9.654808228959270, −9.073724457948595, −8.750184825654707, −8.181004851387852, −7.725059954050977, −7.362419675095385, −6.826341968957307, −6.261616165950053, −5.743186334148395, −5.459708802518489, −4.595515929905403, −4.201524614034940, −3.909253966275782, −3.232035308029930, −2.478828833374619, −2.154863401192117, −1.502292838032422, −0.5809289559902394, 0, 0.5809289559902394, 1.502292838032422, 2.154863401192117, 2.478828833374619, 3.232035308029930, 3.909253966275782, 4.201524614034940, 4.595515929905403, 5.459708802518489, 5.743186334148395, 6.261616165950053, 6.826341968957307, 7.362419675095385, 7.725059954050977, 8.181004851387852, 8.750184825654707, 9.073724457948595, 9.654808228959270, 10.07536175726445, 10.68615837484667, 10.99678754229040, 11.44502604265131, 11.80637252971138, 12.47517697162411, 12.88296072866546

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