Properties

Label 2-44880-1.1-c1-0-63
Degree $2$
Conductor $44880$
Sign $-1$
Analytic cond. $358.368$
Root an. cond. $18.9306$
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 + 4·7-s + 9-s − 11-s + 2·13-s − 15-s − 17-s + 4·19-s − 4·21-s + 25-s − 27-s − 6·29-s − 8·31-s + 33-s + 4·35-s − 10·37-s − 2·39-s + 6·41-s + 4·43-s + 45-s − 12·47-s + 9·49-s + 51-s + 6·53-s − 55-s − 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.258·15-s − 0.242·17-s + 0.917·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.174·33-s + 0.676·35-s − 1.64·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s + 9/7·49-s + 0.140·51-s + 0.824·53-s − 0.134·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44880\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(358.368\)
Root analytic conductor: \(18.9306\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{44880} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 44880,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.00420119191960, −14.29337963047233, −13.91047566244983, −13.46278877047095, −12.71810363060501, −12.41007634138371, −11.59301697717461, −11.21234079393468, −10.95446883040669, −10.33543128489932, −9.736271705844577, −8.995066077789395, −8.737189008276025, −7.774668262688718, −7.610152093739445, −6.920194704615267, −6.168028263298123, −5.566971453534268, −5.184407290507947, −4.714934146502039, −3.901013057638917, −3.315397766043771, −2.250601906345397, −1.690619403824845, −1.148420872714589, 0, 1.148420872714589, 1.690619403824845, 2.250601906345397, 3.315397766043771, 3.901013057638917, 4.714934146502039, 5.184407290507947, 5.566971453534268, 6.168028263298123, 6.920194704615267, 7.610152093739445, 7.774668262688718, 8.737189008276025, 8.995066077789395, 9.736271705844577, 10.33543128489932, 10.95446883040669, 11.21234079393468, 11.59301697717461, 12.41007634138371, 12.71810363060501, 13.46278877047095, 13.91047566244983, 14.29337963047233, 15.00420119191960

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