Properties

Degree $2$
Conductor $18496$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 3·9-s − 6·13-s − 25-s − 10·29-s − 2·37-s − 10·41-s + 6·45-s − 7·49-s − 14·53-s − 10·61-s + 12·65-s + 6·73-s + 9·81-s + 10·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 18·117-s + ⋯
L(s)  = 1  − 0.894·5-s − 9-s − 1.66·13-s − 1/5·25-s − 1.85·29-s − 0.328·37-s − 1.56·41-s + 0.894·45-s − 49-s − 1.92·53-s − 1.28·61-s + 1.48·65-s + 0.702·73-s + 81-s + 1.05·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.66·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18496\)    =    \(2^{6} \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{18496} (1, \cdot )$
Sato-Tate group: $N(\mathrm{U}(1))$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 18496,\ (\ :1/2),\ 1)\)

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)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−16.41139148306097, −15.68014070354751, −15.02347669870304, −14.90065908513053, −14.17514136725227, −13.69716903836960, −12.88724628026910, −12.35317161202679, −11.91512481602633, −11.32344030410068, −11.00013320576722, −10.11380251625983, −9.564009360242546, −9.034817023594925, −8.281440326899827, −7.743271280419343, −7.376789118727272, −6.594085517114712, −5.888286882148094, −5.082418334932807, −4.757960636394168, −3.741160032473476, −3.267286720059900, −2.452663280139293, −1.657392291607183, 0, 0, 1.657392291607183, 2.452663280139293, 3.267286720059900, 3.741160032473476, 4.757960636394168, 5.082418334932807, 5.888286882148094, 6.594085517114712, 7.376789118727272, 7.743271280419343, 8.281440326899827, 9.034817023594925, 9.564009360242546, 10.11380251625983, 11.00013320576722, 11.32344030410068, 11.91512481602633, 12.35317161202679, 12.88724628026910, 13.69716903836960, 14.17514136725227, 14.90065908513053, 15.02347669870304, 15.68014070354751, 16.41139148306097

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