Properties

Label 2-162288-1.1-c1-0-55
Degree $2$
Conductor $162288$
Sign $1$
Analytic cond. $1295.87$
Root an. cond. $35.9982$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 11-s − 4·17-s + 7·19-s + 23-s − 25-s + 4·29-s − 4·31-s + 4·37-s + 9·41-s − 6·43-s − 3·47-s + 11·53-s + 2·55-s + 3·59-s + 13·61-s − 6·67-s + 4·71-s + 8·73-s + 16·79-s + 2·83-s + 8·85-s + 12·89-s − 14·95-s + 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.301·11-s − 0.970·17-s + 1.60·19-s + 0.208·23-s − 1/5·25-s + 0.742·29-s − 0.718·31-s + 0.657·37-s + 1.40·41-s − 0.914·43-s − 0.437·47-s + 1.51·53-s + 0.269·55-s + 0.390·59-s + 1.66·61-s − 0.733·67-s + 0.474·71-s + 0.936·73-s + 1.80·79-s + 0.219·83-s + 0.867·85-s + 1.27·89-s − 1.43·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162288\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1295.87\)
Root analytic conductor: \(35.9982\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{162288} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 162288,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.112172159\)
\(L(\frac12)\) \(\approx\) \(2.112172159\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
23 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−13.22530070849767, −12.88773635599119, −12.17044797095218, −11.74073466793159, −11.55254473561869, −10.85489012137438, −10.60675068500002, −9.803893757095457, −9.475176315248048, −8.970894475050383, −8.250222930620485, −8.071050333953479, −7.299121024035006, −7.181226106051204, −6.436627758283924, −5.911326227145044, −5.158956534609548, −4.940663936117907, −4.131516961530311, −3.755508553587342, −3.186169221255808, −2.511423461009333, −1.979077816842015, −0.9557188506797936, −0.5192675993076544, 0.5192675993076544, 0.9557188506797936, 1.979077816842015, 2.511423461009333, 3.186169221255808, 3.755508553587342, 4.131516961530311, 4.940663936117907, 5.158956534609548, 5.911326227145044, 6.436627758283924, 7.181226106051204, 7.299121024035006, 8.071050333953479, 8.250222930620485, 8.970894475050383, 9.475176315248048, 9.803893757095457, 10.60675068500002, 10.85489012137438, 11.55254473561869, 11.74073466793159, 12.17044797095218, 12.88773635599119, 13.22530070849767

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