Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 59 \cdot 113 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 7-s + 9-s + 2·10-s + 4·11-s − 2·12-s − 4·13-s + 2·14-s + 15-s − 4·16-s + 2·17-s − 2·18-s − 2·20-s + 21-s − 8·22-s − 7·23-s + 25-s + 8·26-s − 27-s − 2·28-s − 6·29-s − 2·30-s − 8·31-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.577·12-s − 1.10·13-s + 0.534·14-s + 0.258·15-s − 16-s + 0.485·17-s − 0.471·18-s − 0.447·20-s + 0.218·21-s − 1.70·22-s − 1.45·23-s + 1/5·25-s + 1.56·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s − 0.365·30-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100005\)    =    \(3 \cdot 5 \cdot 59 \cdot 113\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{100005} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 100005,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;59,\;113\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;59,\;113\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
5 \( 1 + T \)
59 \( 1 - T \)
113 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.35531784025453, −13.86292662999597, −13.11082212247720, −12.52710278025528, −12.16367166396146, −11.65144926239270, −11.28313252012876, −10.72279977632798, −10.15566873713051, −9.668672872282089, −9.517621128468811, −8.823375599317415, −8.317318598305679, −7.783200868260622, −7.226627710353203, −6.898114304772559, −6.423396267049531, −5.616533958778447, −5.153335960748824, −4.406843327480296, −3.767211558134807, −3.385658597216329, −2.172864560437771, −1.816414789089624, −1.070281893583117, 0, 0, 1.070281893583117, 1.816414789089624, 2.172864560437771, 3.385658597216329, 3.767211558134807, 4.406843327480296, 5.153335960748824, 5.616533958778447, 6.423396267049531, 6.898114304772559, 7.226627710353203, 7.783200868260622, 8.317318598305679, 8.823375599317415, 9.517621128468811, 9.668672872282089, 10.15566873713051, 10.72279977632798, 11.28313252012876, 11.65144926239270, 12.16367166396146, 12.52710278025528, 13.11082212247720, 13.86292662999597, 14.35531784025453

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