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Properties

Label 2-38440-1.1-c1-0-5
Degree $2$
Conductor $38440$
Sign $-1$
Analytic cond. $306.944$
Root an. cond. $17.5198$
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 − 3·9-s − 4·11-s + 2·13-s − 2·17-s + 4·19-s − 4·23-s + 25-s + 2·29-s − 4·35-s − 6·37-s − 6·41-s + 8·43-s − 3·45-s + 4·47-s + 9·49-s − 6·53-s − 4·55-s − 4·59-s + 2·61-s + 12·63-s + 2·65-s + 8·67-s + 6·73-s + 16·77-s + 9·81-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s − 9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 0.676·35-s − 0.986·37-s − 0.937·41-s + 1.21·43-s − 0.447·45-s + 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.539·55-s − 0.520·59-s + 0.256·61-s + 1.51·63-s + 0.248·65-s + 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38440\)    =    \(2^{3} \cdot 5 \cdot 31^{2}\)
Sign: $-1$
Analytic conductor: \(306.944\)
Root analytic conductor: \(17.5198\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 38440,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
31 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
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

−15.43815746361261, −14.38754131588926, −13.83604849966535, −13.66442167741201, −13.07197524806926, −12.49112560940015, −12.13277816326098, −11.36149992766384, −10.80219334245605, −10.33312487882299, −9.797249297258832, −9.308261958218674, −8.750990408812642, −8.196902049846805, −7.565522930295079, −6.877428622669713, −6.271363294706188, −5.872855391354083, −5.334063788862475, −4.694043585463828, −3.597458344588844, −3.303222278121751, −2.591225444641873, −2.047915018526450, −0.7663994518965566, 0, 0.7663994518965566, 2.047915018526450, 2.591225444641873, 3.303222278121751, 3.597458344588844, 4.694043585463828, 5.334063788862475, 5.872855391354083, 6.271363294706188, 6.877428622669713, 7.565522930295079, 8.196902049846805, 8.750990408812642, 9.308261958218674, 9.797249297258832, 10.33312487882299, 10.80219334245605, 11.36149992766384, 12.13277816326098, 12.49112560940015, 13.07197524806926, 13.66442167741201, 13.83604849966535, 14.38754131588926, 15.43815746361261

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