Properties

Label 2-47190-1.1-c1-0-5
Degree $2$
Conductor $47190$
Sign $1$
Analytic cond. $376.814$
Root an. cond. $19.4116$
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 + 2·7-s − 8-s + 9-s + 10-s − 12-s − 13-s − 2·14-s + 15-s + 16-s − 4·17-s − 18-s + 19-s − 20-s − 2·21-s + 24-s + 25-s + 26-s − 27-s + 2·28-s − 29-s − 30-s + 8·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.436·21-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s − 0.185·29-s − 0.182·30-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47190\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(376.814\)
Root analytic conductor: \(19.4116\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{47190} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 47190,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.188033469\)
\(L(\frac12)\) \(\approx\) \(1.188033469\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−14.82611066068336, −14.09692656999871, −13.52197848749044, −13.01939487811806, −12.25221907544087, −11.91367751648432, −11.34460653743498, −11.12817132840395, −10.36073276841188, −10.07882435818027, −9.300123125778805, −8.784090930799833, −8.282411997998053, −7.629465433729753, −7.354872060845986, −6.487044316824967, −6.245060296659745, −5.323638595882300, −4.809071015956921, −4.282151816039678, −3.544392944535204, −2.608196096735856, −2.068006387270065, −1.151920219103826, −0.5090037390705537, 0.5090037390705537, 1.151920219103826, 2.068006387270065, 2.608196096735856, 3.544392944535204, 4.282151816039678, 4.809071015956921, 5.323638595882300, 6.245060296659745, 6.487044316824967, 7.354872060845986, 7.629465433729753, 8.282411997998053, 8.784090930799833, 9.300123125778805, 10.07882435818027, 10.36073276841188, 11.12817132840395, 11.34460653743498, 11.91367751648432, 12.25221907544087, 13.01939487811806, 13.52197848749044, 14.09692656999871, 14.82611066068336

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