Properties

Label 2-22848-1.1-c1-0-37
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 − 5·11-s + 5·13-s + 15-s + 17-s − 5·19-s + 21-s + 23-s − 4·25-s − 27-s + 6·29-s + 6·31-s + 5·33-s + 35-s − 4·37-s − 5·39-s + 7·41-s − 7·43-s − 45-s − 6·47-s + 49-s − 51-s − 6·53-s + 5·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.38·13-s + 0.258·15-s + 0.242·17-s − 1.14·19-s + 0.218·21-s + 0.208·23-s − 4/5·25-s − 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.870·33-s + 0.169·35-s − 0.657·37-s − 0.800·39-s + 1.09·41-s − 1.06·43-s − 0.149·45-s − 0.875·47-s + 1/7·49-s − 0.140·51-s − 0.824·53-s + 0.674·55-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: Trivial
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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−15.80857506947034, −15.46979699711997, −14.82343487084406, −14.04505955645753, −13.37816410545009, −13.09382130831707, −12.58734233104592, −11.83415972623794, −11.47531781180809, −10.70492627878955, −10.41859327474591, −9.941108168600991, −9.047029748643226, −8.327126755465634, −8.081722660350342, −7.358389230491419, −6.470997388356633, −6.254346949191078, −5.472446709052511, −4.838591779949009, −4.198589283601330, −3.470243095012436, −2.795840713946795, −1.911363528150883, −0.8627079721000353, 0, 0.8627079721000353, 1.911363528150883, 2.795840713946795, 3.470243095012436, 4.198589283601330, 4.838591779949009, 5.472446709052511, 6.254346949191078, 6.470997388356633, 7.358389230491419, 8.081722660350342, 8.327126755465634, 9.047029748643226, 9.941108168600991, 10.41859327474591, 10.70492627878955, 11.47531781180809, 11.83415972623794, 12.58734233104592, 13.09382130831707, 13.37816410545009, 14.04505955645753, 14.82343487084406, 15.46979699711997, 15.80857506947034

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