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Properties

Label 2-32487-1.1-c1-0-13
Degree $2$
Conductor $32487$
Sign $-1$
Analytic cond. $259.410$
Root an. cond. $16.1062$
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  − 3-s − 2·4-s + 5-s + 9-s + 6·11-s + 2·12-s − 13-s − 15-s + 4·16-s + 17-s + 7·19-s − 2·20-s − 3·23-s − 4·25-s − 27-s − 7·29-s + 7·31-s − 6·33-s − 2·36-s + 8·37-s + 39-s + 2·41-s − 9·43-s − 12·44-s + 45-s + 7·47-s − 4·48-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.447·5-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 0.277·13-s − 0.258·15-s + 16-s + 0.242·17-s + 1.60·19-s − 0.447·20-s − 0.625·23-s − 4/5·25-s − 0.192·27-s − 1.29·29-s + 1.25·31-s − 1.04·33-s − 1/3·36-s + 1.31·37-s + 0.160·39-s + 0.312·41-s − 1.37·43-s − 1.80·44-s + 0.149·45-s + 1.02·47-s − 0.577·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32487\)    =    \(3 \cdot 7^{2} \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(259.410\)
Root analytic conductor: \(16.1062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 32487,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
7 \( 1 \)
13 \( 1 + T \)
17 \( 1 - T \)
good2 \( 1 + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−15.15474145831566, −14.68527563746090, −14.14803061473689, −13.71761447024007, −13.37176182116052, −12.58966646371179, −12.08816558280531, −11.63324913329783, −11.28475362431317, −10.24924591774938, −9.851242805826774, −9.420259950753469, −9.123841447298788, −8.245238831385604, −7.696874033088170, −7.076763980102911, −6.297977073027051, −5.865015708109562, −5.337813237102919, −4.642140783363455, −3.992954623078385, −3.615792909342972, −2.633789577324004, −1.450803114138502, −1.120717401808596, 0, 1.120717401808596, 1.450803114138502, 2.633789577324004, 3.615792909342972, 3.992954623078385, 4.642140783363455, 5.337813237102919, 5.865015708109562, 6.297977073027051, 7.076763980102911, 7.696874033088170, 8.245238831385604, 9.123841447298788, 9.420259950753469, 9.851242805826774, 10.24924591774938, 11.28475362431317, 11.63324913329783, 12.08816558280531, 12.58966646371179, 13.37176182116052, 13.71761447024007, 14.14803061473689, 14.68527563746090, 15.15474145831566

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