Properties

Label 2-27930-1.1-c1-0-51
Degree $2$
Conductor $27930$
Sign $1$
Analytic cond. $223.022$
Root an. cond. $14.9339$
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  + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 6·11-s + 12-s + 15-s + 16-s + 3·17-s + 18-s − 19-s + 20-s + 6·22-s − 6·23-s + 24-s + 25-s + 27-s − 8·29-s + 30-s − 8·31-s + 32-s + 6·33-s + 3·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s + 0.288·12-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s + 1.27·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 1.48·29-s + 0.182·30-s − 1.43·31-s + 0.176·32-s + 1.04·33-s + 0.514·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27930\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(223.022\)
Root analytic conductor: \(14.9339\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{27930} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 27930,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.331154531\)
\(L(\frac12)\) \(\approx\) \(6.331154531\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
19 \( 1 + T \)
good11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−15.16066548342014, −14.42071229078035, −14.17687700263135, −13.95582338609930, −12.99595555080130, −12.72080375175061, −12.15239586020316, −11.48327501291953, −11.13230990037481, −10.33517474501100, −9.579906003671093, −9.437532979799488, −8.712884971950685, −8.009594544471712, −7.412984701257154, −6.823367267008177, −6.192873390842120, −5.747475335229164, −5.027292997039162, −4.145504686321913, −3.739541591122202, −3.288133228826715, −2.064933634158853, −1.886576666580240, −0.8560975114809491, 0.8560975114809491, 1.886576666580240, 2.064933634158853, 3.288133228826715, 3.739541591122202, 4.145504686321913, 5.027292997039162, 5.747475335229164, 6.192873390842120, 6.823367267008177, 7.412984701257154, 8.009594544471712, 8.712884971950685, 9.437532979799488, 9.579906003671093, 10.33517474501100, 11.13230990037481, 11.48327501291953, 12.15239586020316, 12.72080375175061, 12.99595555080130, 13.95582338609930, 14.17687700263135, 14.42071229078035, 15.16066548342014

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