Properties

Label 2-29645-1.1-c1-0-7
Degree $2$
Conductor $29645$
Sign $-1$
Analytic cond. $236.716$
Root an. cond. $15.3855$
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  + 2-s − 4-s + 5-s − 3·8-s − 3·9-s + 10-s − 16-s − 6·17-s − 3·18-s − 2·19-s − 20-s − 5·23-s + 25-s + 29-s + 8·31-s + 5·32-s − 6·34-s + 3·36-s + 6·37-s − 2·38-s − 3·40-s + 7·41-s + 5·43-s − 3·45-s − 5·46-s − 47-s + 50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.06·8-s − 9-s + 0.316·10-s − 1/4·16-s − 1.45·17-s − 0.707·18-s − 0.458·19-s − 0.223·20-s − 1.04·23-s + 1/5·25-s + 0.185·29-s + 1.43·31-s + 0.883·32-s − 1.02·34-s + 1/2·36-s + 0.986·37-s − 0.324·38-s − 0.474·40-s + 1.09·41-s + 0.762·43-s − 0.447·45-s − 0.737·46-s − 0.145·47-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29645\)    =    \(5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(236.716\)
Root analytic conductor: \(15.3855\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{29645} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29645,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−15.20357425126103, −14.79848213347382, −14.21425714054707, −13.78499932524452, −13.50015257712894, −12.78775183909694, −12.40292037174006, −11.78224360306930, −11.13905532834812, −10.81348667528871, −9.821499344206971, −9.534255051824255, −8.909113848038572, −8.221148167836746, −8.094290500508474, −6.820383215142794, −6.398924706217686, −5.853204122208761, −5.373905759329680, −4.528468860301801, −4.249303450455200, −3.440850859586849, −2.503749193517600, −2.341747911448824, −0.9050841421708072, 0, 0.9050841421708072, 2.341747911448824, 2.503749193517600, 3.440850859586849, 4.249303450455200, 4.528468860301801, 5.373905759329680, 5.853204122208761, 6.398924706217686, 6.820383215142794, 8.094290500508474, 8.221148167836746, 8.909113848038572, 9.534255051824255, 9.821499344206971, 10.81348667528871, 11.13905532834812, 11.78224360306930, 12.40292037174006, 12.78775183909694, 13.50015257712894, 13.78499932524452, 14.21425714054707, 14.79848213347382, 15.20357425126103

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