Properties

Label 2-8880-1.1-c1-0-24
Degree $2$
Conductor $8880$
Sign $1$
Analytic cond. $70.9071$
Root an. cond. $8.42063$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 4·7-s + 9-s − 6·11-s + 2·13-s − 15-s − 6·17-s − 2·19-s − 4·21-s + 25-s − 27-s + 6·29-s − 8·31-s + 6·33-s + 4·35-s + 37-s − 2·39-s − 6·41-s − 8·43-s + 45-s − 6·47-s + 9·49-s + 6·51-s + 6·53-s − 6·55-s + 2·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.458·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 1.04·33-s + 0.676·35-s + 0.164·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s − 0.875·47-s + 9/7·49-s + 0.840·51-s + 0.824·53-s − 0.809·55-s + 0.264·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8880\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(70.9071\)
Root analytic conductor: \(8.42063\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8880} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8880,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.677633988\)
\(L(\frac12)\) \(\approx\) \(1.677633988\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
37 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−7.88490674930624237986040718133, −6.96165291772688103919348452577, −6.42281917887630762159396976932, −5.40209409240129424016558382093, −5.10773611823564971575397049931, −4.54819374963735396242901787561, −3.53320746341258620082863360917, −2.23988396323307688436465491180, −1.95351103406954527954573674898, −0.63181959746510107924075287826, 0.63181959746510107924075287826, 1.95351103406954527954573674898, 2.23988396323307688436465491180, 3.53320746341258620082863360917, 4.54819374963735396242901787561, 5.10773611823564971575397049931, 5.40209409240129424016558382093, 6.42281917887630762159396976932, 6.96165291772688103919348452577, 7.88490674930624237986040718133

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