Properties

Label 2-29645-1.1-c1-0-4
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 5-s − 3·8-s − 3·9-s − 10-s − 6·13-s − 16-s + 6·17-s − 3·18-s − 4·19-s + 20-s − 8·23-s + 25-s − 6·26-s + 10·29-s + 4·31-s + 5·32-s + 6·34-s + 3·36-s + 6·37-s − 4·38-s + 3·40-s − 10·41-s − 4·43-s + 3·45-s − 8·46-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 − 1.66·13-s − 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.917·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s − 1.17·26-s + 1.85·29-s + 0.718·31-s + 0.883·32-s + 1.02·34-s + 1/2·36-s + 0.986·37-s − 0.648·38-s + 0.474·40-s − 1.56·41-s − 0.609·43-s + 0.447·45-s − 1.17·46-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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.16679635115742, −14.78653416904217, −14.33274430915070, −13.88583794602475, −13.52156424934345, −12.49323767680900, −12.25358362239825, −11.98769193318958, −11.44616630451718, −10.44350152859630, −9.998623646906526, −9.646133534228225, −8.738961598926740, −8.242325671950773, −7.950345171396573, −7.121388920573685, −6.262531885910062, −5.964521350824423, −5.020235200875935, −4.870383582021150, −4.065475060076043, −3.423391816250314, −2.776624652695021, −2.208886948047193, −0.7770892604278816, 0, 0.7770892604278816, 2.208886948047193, 2.776624652695021, 3.423391816250314, 4.065475060076043, 4.870383582021150, 5.020235200875935, 5.964521350824423, 6.262531885910062, 7.121388920573685, 7.950345171396573, 8.242325671950773, 8.738961598926740, 9.646133534228225, 9.998623646906526, 10.44350152859630, 11.44616630451718, 11.98769193318958, 12.25358362239825, 12.49323767680900, 13.52156424934345, 13.88583794602475, 14.33274430915070, 14.78653416904217, 15.16679635115742

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