Properties

Label 2-12882-1.1-c1-0-7
Degree $2$
Conductor $12882$
Sign $-1$
Analytic cond. $102.863$
Root an. cond. $10.1421$
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  − 2-s − 3-s + 4-s + 4·5-s + 6-s − 4·7-s − 8-s + 9-s − 4·10-s − 12-s + 4·14-s − 4·15-s + 16-s − 6·17-s − 18-s + 19-s + 4·20-s + 4·21-s + 6·23-s + 24-s + 11·25-s − 27-s − 4·28-s − 6·29-s + 4·30-s + 10·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.288·12-s + 1.06·14-s − 1.03·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.229·19-s + 0.894·20-s + 0.872·21-s + 1.25·23-s + 0.204·24-s + 11/5·25-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + 0.730·30-s + 1.79·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12882\)    =    \(2 \cdot 3 \cdot 19 \cdot 113\)
Sign: $-1$
Analytic conductor: \(102.863\)
Root analytic conductor: \(10.1421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{12882} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 12882,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 + T \)
19 \( 1 - T \)
113 \( 1 - T \)
good5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−16.71057692986472, −16.17062216510382, −15.59317270232023, −15.08259308484567, −14.19292412022844, −13.45882883913358, −13.21962386723922, −12.73292644138163, −12.00567540508149, −11.23111606754041, −10.51154304758590, −10.27649765417188, −9.599995787581816, −9.038728106102049, −8.924632692064198, −7.636579962842694, −6.728617772844334, −6.582336018165581, −6.050929062842574, −5.332607180305933, −4.649022605793482, −3.413360827621125, −2.706198156886974, −2.009209144097549, −1.095993255036032, 0, 1.095993255036032, 2.009209144097549, 2.706198156886974, 3.413360827621125, 4.649022605793482, 5.332607180305933, 6.050929062842574, 6.582336018165581, 6.728617772844334, 7.636579962842694, 8.924632692064198, 9.038728106102049, 9.599995787581816, 10.27649765417188, 10.51154304758590, 11.23111606754041, 12.00567540508149, 12.73292644138163, 13.21962386723922, 13.45882883913358, 14.19292412022844, 15.08259308484567, 15.59317270232023, 16.17062216510382, 16.71057692986472

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