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Properties

Label 2-33640-1.1-c1-0-8
Degree $2$
Conductor $33640$
Sign $-1$
Analytic cond. $268.616$
Root an. cond. $16.3895$
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 − 3·9-s − 2·13-s + 6·17-s + 8·19-s + 25-s − 4·31-s + 2·37-s + 6·41-s − 8·43-s − 3·45-s − 4·47-s − 7·49-s − 10·53-s + 4·59-s + 2·61-s − 2·65-s − 12·67-s − 8·71-s − 2·73-s − 4·79-s + 9·81-s + 12·83-s + 6·85-s + 6·89-s + 8·95-s − 10·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 9-s − 0.554·13-s + 1.45·17-s + 1.83·19-s + 1/5·25-s − 0.718·31-s + 0.328·37-s + 0.937·41-s − 1.21·43-s − 0.447·45-s − 0.583·47-s − 49-s − 1.37·53-s + 0.520·59-s + 0.256·61-s − 0.248·65-s − 1.46·67-s − 0.949·71-s − 0.234·73-s − 0.450·79-s + 81-s + 1.31·83-s + 0.650·85-s + 0.635·89-s + 0.820·95-s − 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33640\)    =    \(2^{3} \cdot 5 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(268.616\)
Root analytic conductor: \(16.3895\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 33640,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
29 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.93452323768348, −14.74836144915352, −14.21707231309148, −13.80099634407065, −13.22235039464020, −12.58656798173206, −12.02450416652179, −11.60725108552534, −11.13310989586707, −10.36573697790383, −9.846833246146013, −9.425151890936158, −8.946899550848471, −8.047671318801228, −7.764337911493542, −7.160837006065696, −6.346563673237365, −5.811691197282501, −5.227376843983599, −4.936907233448792, −3.829846008306035, −3.075256148351655, −2.867865715156110, −1.762061307913531, −1.082719464209456, 0, 1.082719464209456, 1.762061307913531, 2.867865715156110, 3.075256148351655, 3.829846008306035, 4.936907233448792, 5.227376843983599, 5.811691197282501, 6.346563673237365, 7.160837006065696, 7.764337911493542, 8.047671318801228, 8.946899550848471, 9.425151890936158, 9.846833246146013, 10.36573697790383, 11.13310989586707, 11.60725108552534, 12.02450416652179, 12.58656798173206, 13.22235039464020, 13.80099634407065, 14.21707231309148, 14.74836144915352, 14.93452323768348

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