Properties

Degree $2$
Conductor $163170$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s − 4·11-s + 2·13-s + 16-s + 2·17-s − 4·19-s + 20-s + 4·22-s + 8·23-s + 25-s − 2·26-s + 2·29-s − 8·31-s − 32-s − 2·34-s + 37-s + 4·38-s − 40-s + 10·41-s + 12·43-s − 4·44-s − 8·46-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.392·26-s + 0.371·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s + 0.164·37-s + 0.648·38-s − 0.158·40-s + 1.56·41-s + 1.82·43-s − 0.603·44-s − 1.17·46-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(163170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 37\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{163170} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 163170,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
37 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.26413609283194, −12.99609337873068, −12.65475127272625, −12.15956370138957, −11.27063415758647, −11.00238038576766, −10.67401189482640, −10.26446934518221, −9.512369378681659, −9.286306057854770, −8.685047071515336, −8.306216487171595, −7.647818156047886, −7.327778461316447, −6.766825882618436, −6.142565260607807, −5.631185657416643, −5.300426593209351, −4.556334364976770, −3.951364180315047, −3.206158747272081, −2.616869645477461, −2.275789694289695, −1.385085079589816, −0.8612578122161018, 0, 0.8612578122161018, 1.385085079589816, 2.275789694289695, 2.616869645477461, 3.206158747272081, 3.951364180315047, 4.556334364976770, 5.300426593209351, 5.631185657416643, 6.142565260607807, 6.766825882618436, 7.327778461316447, 7.647818156047886, 8.306216487171595, 8.685047071515336, 9.286306057854770, 9.512369378681659, 10.26446934518221, 10.67401189482640, 11.00238038576766, 11.27063415758647, 12.15956370138957, 12.65475127272625, 12.99609337873068, 13.26413609283194

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