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Properties

Label 2-244800-1.1-c1-0-225
Degree $2$
Conductor $244800$
Sign $-1$
Analytic cond. $1954.73$
Root an. cond. $44.2124$
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  − 4·11-s − 2·13-s + 17-s − 4·19-s − 10·29-s + 8·31-s − 2·37-s − 10·41-s + 12·43-s − 7·49-s − 6·53-s + 12·59-s + 10·61-s − 12·67-s − 10·73-s − 8·79-s − 4·83-s + 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.20·11-s − 0.554·13-s + 0.242·17-s − 0.917·19-s − 1.85·29-s + 1.43·31-s − 0.328·37-s − 1.56·41-s + 1.82·43-s − 49-s − 0.824·53-s + 1.56·59-s + 1.28·61-s − 1.46·67-s − 1.17·73-s − 0.900·79-s − 0.439·83-s + 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(244800\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(1954.73\)
Root analytic conductor: \(44.2124\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 244800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
17 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.99566229688687, −12.88496339987523, −12.15443229088822, −11.70094742933712, −11.31574222467888, −10.62585427050169, −10.43488219974738, −9.819761208656495, −9.566878901725527, −8.701257003365823, −8.528294504462651, −7.900295856794826, −7.412926507217181, −7.130292421533244, −6.365487615492384, −5.950154905061894, −5.380872861696427, −4.957342709797696, −4.423127136170162, −3.869508308322296, −3.189775343389660, −2.697040256840693, −2.120297236742076, −1.613583590369521, −0.6225288146000435, 0, 0.6225288146000435, 1.613583590369521, 2.120297236742076, 2.697040256840693, 3.189775343389660, 3.869508308322296, 4.423127136170162, 4.957342709797696, 5.380872861696427, 5.950154905061894, 6.365487615492384, 7.130292421533244, 7.412926507217181, 7.900295856794826, 8.528294504462651, 8.701257003365823, 9.566878901725527, 9.819761208656495, 10.43488219974738, 10.62585427050169, 11.31574222467888, 11.70094742933712, 12.15443229088822, 12.88496339987523, 12.99566229688687

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