Properties

Label 2-8880-1.1-c1-0-91
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 + 3·7-s + 9-s + 5·11-s − 2·13-s + 15-s + 3·17-s + 6·19-s + 3·21-s + 4·23-s + 25-s + 27-s − 29-s + 3·31-s + 5·33-s + 3·35-s − 37-s − 2·39-s − 7·41-s − 3·43-s + 45-s + 2·49-s + 3·51-s + 5·53-s + 5·55-s + 6·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 1.50·11-s − 0.554·13-s + 0.258·15-s + 0.727·17-s + 1.37·19-s + 0.654·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.185·29-s + 0.538·31-s + 0.870·33-s + 0.507·35-s − 0.164·37-s − 0.320·39-s − 1.09·41-s − 0.457·43-s + 0.149·45-s + 2/7·49-s + 0.420·51-s + 0.686·53-s + 0.674·55-s + 0.794·57-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

\(L(1)\) \(\approx\) \(4.235982944\)
\(L(\frac12)\) \(\approx\) \(4.235982944\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−7.81711342495898736919825913075, −7.05907534135576115476823913226, −6.59491262730150387108087343780, −5.45332600814170740427719090832, −5.08037689340791497885841257552, −4.19501563431572657790163216264, −3.44396822862779270175469256875, −2.63538360512994690563852596523, −1.57580795471895639405087761350, −1.13754192052773864602260253573, 1.13754192052773864602260253573, 1.57580795471895639405087761350, 2.63538360512994690563852596523, 3.44396822862779270175469256875, 4.19501563431572657790163216264, 5.08037689340791497885841257552, 5.45332600814170740427719090832, 6.59491262730150387108087343780, 7.05907534135576115476823913226, 7.81711342495898736919825913075

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