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Properties

Label 2-29645-1.1-c1-0-15
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  + 2·2-s + 3·3-s + 2·4-s − 5-s + 6·6-s + 6·9-s − 2·10-s + 6·12-s − 3·13-s − 3·15-s − 4·16-s + 3·17-s + 12·18-s − 6·19-s − 2·20-s − 4·23-s + 25-s − 6·26-s + 9·27-s + 29-s − 6·30-s + 6·31-s − 8·32-s + 6·34-s + 12·36-s − 12·38-s − 9·39-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s + 4-s − 0.447·5-s + 2.44·6-s + 2·9-s − 0.632·10-s + 1.73·12-s − 0.832·13-s − 0.774·15-s − 16-s + 0.727·17-s + 2.82·18-s − 1.37·19-s − 0.447·20-s − 0.834·23-s + 1/5·25-s − 1.17·26-s + 1.73·27-s + 0.185·29-s − 1.09·30-s + 1.07·31-s − 1.41·32-s + 1.02·34-s + 2·36-s − 1.94·38-s − 1.44·39-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: Trivial
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 \( 1 - p T + p T^{2} \)
3 \( 1 - p T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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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

−15.08754549056891, −14.83054656993210, −14.20195178623707, −14.08410747148585, −13.36134095684337, −12.92204046249730, −12.44765644513976, −12.00354713294689, −11.41210844964827, −10.48242799798425, −10.02583200417301, −9.414054107623108, −8.756176519429601, −8.307065795285604, −7.713032310620319, −7.235852335676212, −6.453219016282172, −5.979986989305075, −4.944237359617351, −4.453605141357178, −4.088718920282792, −3.225216479138044, −2.999035496484020, −2.246345811283511, −1.588858943809882, 0, 1.588858943809882, 2.246345811283511, 2.999035496484020, 3.225216479138044, 4.088718920282792, 4.453605141357178, 4.944237359617351, 5.979986989305075, 6.453219016282172, 7.235852335676212, 7.713032310620319, 8.307065795285604, 8.756176519429601, 9.414054107623108, 10.02583200417301, 10.48242799798425, 11.41210844964827, 12.00354713294689, 12.44765644513976, 12.92204046249730, 13.36134095684337, 14.08410747148585, 14.20195178623707, 14.83054656993210, 15.08754549056891

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