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Properties

Label 2-30528-1.1-c1-0-45
Degree $2$
Conductor $30528$
Sign $-1$
Analytic cond. $243.767$
Root an. cond. $15.6130$
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  − 3·5-s + 4·7-s + 3·11-s + 2·13-s + 7·17-s − 6·19-s − 5·23-s + 4·25-s + 8·29-s + 31-s − 12·35-s − 2·37-s + 12·41-s − 3·43-s − 10·47-s + 9·49-s + 53-s − 9·55-s − 11·59-s − 14·61-s − 6·65-s − 14·67-s − 8·71-s − 6·73-s + 12·77-s − 79-s + 14·83-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.51·7-s + 0.904·11-s + 0.554·13-s + 1.69·17-s − 1.37·19-s − 1.04·23-s + 4/5·25-s + 1.48·29-s + 0.179·31-s − 2.02·35-s − 0.328·37-s + 1.87·41-s − 0.457·43-s − 1.45·47-s + 9/7·49-s + 0.137·53-s − 1.21·55-s − 1.43·59-s − 1.79·61-s − 0.744·65-s − 1.71·67-s − 0.949·71-s − 0.702·73-s + 1.36·77-s − 0.112·79-s + 1.53·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30528\)    =    \(2^{6} \cdot 3^{2} \cdot 53\)
Sign: $-1$
Analytic conductor: \(243.767\)
Root analytic conductor: \(15.6130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 30528,\ (\ :1/2),\ -1)\)

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 \( 1 \)
53 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−15.16718286381881, −14.85864120687573, −14.44135796418755, −13.99989592794308, −13.35327078892405, −12.34488627378234, −12.08720442926051, −11.84115748975912, −11.12343676300835, −10.70544038216095, −10.20335822899287, −9.310069994409294, −8.647208927098833, −8.240868204811100, −7.707852229219986, −7.522613025702962, −6.354721457708292, −6.135081961440615, −5.119823097631681, −4.473265846292449, −4.160230520517510, −3.499223960257440, −2.717087675107582, −1.562125267322236, −1.204872638855496, 0, 1.204872638855496, 1.562125267322236, 2.717087675107582, 3.499223960257440, 4.160230520517510, 4.473265846292449, 5.119823097631681, 6.135081961440615, 6.354721457708292, 7.522613025702962, 7.707852229219986, 8.240868204811100, 8.647208927098833, 9.310069994409294, 10.20335822899287, 10.70544038216095, 11.12343676300835, 11.84115748975912, 12.08720442926051, 12.34488627378234, 13.35327078892405, 13.99989592794308, 14.44135796418755, 14.85864120687573, 15.16718286381881

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