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Properties

Label 2-379050-1.1-c1-0-123
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 − 2·11-s + 12-s + 14-s + 16-s − 8·17-s − 18-s − 21-s + 2·22-s + 6·23-s − 24-s + 27-s − 28-s + 2·29-s + 8·31-s − 32-s − 2·33-s + 8·34-s + 36-s − 10·37-s − 2·41-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 − 0.603·11-s + 0.288·12-s + 0.267·14-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 0.218·21-s + 0.426·22-s + 1.25·23-s − 0.204·24-s + 0.192·27-s − 0.188·28-s + 0.371·29-s + 1.43·31-s − 0.176·32-s − 0.348·33-s + 1.37·34-s + 1/6·36-s − 1.64·37-s − 0.312·41-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−12.76803628446188, −12.22216613281667, −11.77147095784794, −11.21647462833252, −10.74762594152490, −10.46293691408564, −10.00044419941299, −9.395087060285303, −9.012565960706066, −8.719836862807962, −8.259273902975232, −7.764918465026252, −7.159770864647808, −6.901756515031447, −6.329027725558609, −6.009134273442088, −5.066153183817574, −4.765689473279394, −4.266110189525722, −3.459375254533351, −3.027045160332571, −2.571565832310632, −2.038968807113454, −1.449525509450846, −0.6609524439241565, 0, 0.6609524439241565, 1.449525509450846, 2.038968807113454, 2.571565832310632, 3.027045160332571, 3.459375254533351, 4.266110189525722, 4.765689473279394, 5.066153183817574, 6.009134273442088, 6.329027725558609, 6.901756515031447, 7.159770864647808, 7.764918465026252, 8.259273902975232, 8.719836862807962, 9.012565960706066, 9.395087060285303, 10.00044419941299, 10.46293691408564, 10.74762594152490, 11.21647462833252, 11.77147095784794, 12.22216613281667, 12.76803628446188

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