Properties

Label 2-22848-1.1-c1-0-75
Degree $2$
Conductor $22848$
Sign $-1$
Analytic cond. $182.442$
Root an. cond. $13.5071$
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 + 2·5-s − 7-s + 9-s + 6·13-s + 2·15-s + 17-s − 21-s − 8·23-s − 25-s + 27-s + 6·29-s − 8·31-s − 2·35-s − 10·37-s + 6·39-s − 6·41-s − 12·43-s + 2·45-s + 49-s + 51-s + 10·53-s + 8·59-s − 6·61-s − 63-s + 12·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.516·15-s + 0.242·17-s − 0.218·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.338·35-s − 1.64·37-s + 0.960·39-s − 0.937·41-s − 1.82·43-s + 0.298·45-s + 1/7·49-s + 0.140·51-s + 1.37·53-s + 1.04·59-s − 0.768·61-s − 0.125·63-s + 1.48·65-s − 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22848\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(182.442\)
Root analytic conductor: \(13.5071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{22848} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 22848,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.79143997149466, −15.27122220415730, −14.53598319707156, −14.01429978014087, −13.57949287274309, −13.31171001104949, −12.66321593643214, −11.89143080134049, −11.55450228143933, −10.44757105212995, −10.31542397638784, −9.802754342701026, −8.963891006076287, −8.621460233465600, −8.154376577888664, −7.258165403246340, −6.703235744523883, −5.999580912592519, −5.690550836835603, −4.844200458800341, −3.815738986514928, −3.590404087911788, −2.714341885688298, −1.781430607635352, −1.449857103120128, 0, 1.449857103120128, 1.781430607635352, 2.714341885688298, 3.590404087911788, 3.815738986514928, 4.844200458800341, 5.690550836835603, 5.999580912592519, 6.703235744523883, 7.258165403246340, 8.154376577888664, 8.621460233465600, 8.963891006076287, 9.802754342701026, 10.31542397638784, 10.44757105212995, 11.55450228143933, 11.89143080134049, 12.66321593643214, 13.31171001104949, 13.57949287274309, 14.01429978014087, 14.53598319707156, 15.27122220415730, 15.79143997149466

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