Properties

Label 2-22848-1.1-c1-0-47
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 − 5-s − 7-s + 9-s − 3·11-s − 3·13-s − 15-s + 17-s − 3·19-s − 21-s + 7·23-s − 4·25-s + 27-s + 6·29-s + 10·31-s − 3·33-s + 35-s − 4·37-s − 3·39-s − 9·41-s − 9·43-s − 45-s + 6·47-s + 49-s + 51-s + 10·53-s + 3·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.832·13-s − 0.258·15-s + 0.242·17-s − 0.688·19-s − 0.218·21-s + 1.45·23-s − 4/5·25-s + 0.192·27-s + 1.11·29-s + 1.79·31-s − 0.522·33-s + 0.169·35-s − 0.657·37-s − 0.480·39-s − 1.40·41-s − 1.37·43-s − 0.149·45-s + 0.875·47-s + 1/7·49-s + 0.140·51-s + 1.37·53-s + 0.404·55-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 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.50630589277525, −15.29467805689539, −14.89235848110905, −13.97722750511358, −13.69211885831164, −13.05810079428831, −12.58641112573371, −11.94346943361960, −11.57811766499747, −10.65178786886937, −10.08269695112832, −9.961774321800147, −8.969923711197601, −8.455058638360948, −8.073158521153884, −7.308642367417982, −6.873182738699420, −6.246706025019926, −5.212170384384418, −4.887201099453239, −4.093801269317643, −3.331227341721045, −2.743088067088669, −2.177952361718039, −1.008018720802396, 0, 1.008018720802396, 2.177952361718039, 2.743088067088669, 3.331227341721045, 4.093801269317643, 4.887201099453239, 5.212170384384418, 6.246706025019926, 6.873182738699420, 7.308642367417982, 8.073158521153884, 8.455058638360948, 8.969923711197601, 9.961774321800147, 10.08269695112832, 10.65178786886937, 11.57811766499747, 11.94346943361960, 12.58641112573371, 13.05810079428831, 13.69211885831164, 13.97722750511358, 14.89235848110905, 15.29467805689539, 15.50630589277525

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