Properties

Label 2-29645-1.1-c1-0-6
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 − 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 \( 1 - 8 T + p T^{2} \)
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

−15.28441905088085, −14.64026709273673, −14.30451481329737, −14.02999646809330, −13.33456066186034, −12.74926081815842, −12.21441857525384, −11.88168698744638, −11.32670883183177, −10.68823976541411, −9.949318240163824, −9.452606284060142, −8.909892151293571, −8.251661404346399, −7.818746109333242, −7.243219976398844, −6.159879410591129, −5.926984185972009, −5.259885935987208, −4.759716992620125, −3.951167403797119, −3.390464988523302, −3.011892037704651, −2.018421100560181, −0.8694724000746957, 0, 0.8694724000746957, 2.018421100560181, 3.011892037704651, 3.390464988523302, 3.951167403797119, 4.759716992620125, 5.259885935987208, 5.926984185972009, 6.159879410591129, 7.243219976398844, 7.818746109333242, 8.251661404346399, 8.909892151293571, 9.452606284060142, 9.949318240163824, 10.68823976541411, 11.32670883183177, 11.88168698744638, 12.21441857525384, 12.74926081815842, 13.33456066186034, 14.02999646809330, 14.30451481329737, 14.64026709273673, 15.28441905088085

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