Properties

Label 2-379050-1.1-c1-0-46
Degree $2$
Conductor $379050$
Sign $-1$
Analytic cond. $3026.72$
Root an. cond. $55.0157$
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 + 6-s − 7-s − 8-s + 9-s − 5·11-s − 12-s + 13-s + 14-s + 16-s − 3·17-s − 18-s + 21-s + 5·22-s − 23-s + 24-s − 26-s − 27-s − 28-s + 29-s + 2·31-s − 32-s + 5·33-s + 3·34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.218·21-s + 1.06·22-s − 0.208·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.185·29-s + 0.359·31-s − 0.176·32-s + 0.870·33-s + 0.514·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(379050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(3026.72\)
Root analytic conductor: \(55.0157\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
19 \( 1 \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−12.60387315105168, −12.14517202666243, −11.86372058649876, −11.01801503426687, −10.93462822559450, −10.55222404253607, −9.948103173243336, −9.685123142865427, −9.124637192323382, −8.568067583658594, −8.124791352674542, −7.765964352640649, −7.216875119380225, −6.697582457410390, −6.340742210971913, −5.782139286620926, −5.368721539369961, −4.732029931206766, −4.399191387751227, −3.586132799979709, −2.898261647799114, −2.723465910154997, −1.759550668038256, −1.480932309985668, −0.4350303925098992, 0, 0.4350303925098992, 1.480932309985668, 1.759550668038256, 2.723465910154997, 2.898261647799114, 3.586132799979709, 4.399191387751227, 4.732029931206766, 5.368721539369961, 5.782139286620926, 6.340742210971913, 6.697582457410390, 7.216875119380225, 7.765964352640649, 8.124791352674542, 8.568067583658594, 9.124637192323382, 9.685123142865427, 9.948103173243336, 10.55222404253607, 10.93462822559450, 11.01801503426687, 11.86372058649876, 12.14517202666243, 12.60387315105168

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