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Properties

Label 2-379050-1.1-c1-0-10
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 $0$

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 − 11-s − 12-s + 5·13-s − 14-s + 16-s − 7·17-s − 18-s − 21-s + 22-s − 4·23-s + 24-s − 5·26-s − 27-s + 28-s − 6·29-s − 2·31-s − 32-s + 33-s + 7·34-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 − 0.301·11-s − 0.288·12-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 1.69·17-s − 0.235·18-s − 0.218·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.359·31-s − 0.176·32-s + 0.174·33-s + 1.20·34-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: \(0\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5747823993\)
\(L(\frac12)\) \(\approx\) \(0.5747823993\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
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

−12.49967909400553, −11.83796867829885, −11.30504016258660, −11.08762968680614, −10.84278713858447, −10.38659977406454, −9.627707311734995, −9.338521154920514, −8.901290977906053, −8.385338206448378, −7.920983408878488, −7.526559743234446, −7.005660693973491, −6.361113575469790, −6.079976706136551, −5.731751967611481, −5.015868254158664, −4.422148164742431, −4.031779620501252, −3.498306845066008, −2.649525470426642, −2.192060753519877, −1.583045714829551, −1.076740552587726, −0.2448708019845296, 0.2448708019845296, 1.076740552587726, 1.583045714829551, 2.192060753519877, 2.649525470426642, 3.498306845066008, 4.031779620501252, 4.422148164742431, 5.015868254158664, 5.731751967611481, 6.079976706136551, 6.361113575469790, 7.005660693973491, 7.526559743234446, 7.920983408878488, 8.385338206448378, 8.901290977906053, 9.338521154920514, 9.627707311734995, 10.38659977406454, 10.84278713858447, 11.08762968680614, 11.30504016258660, 11.83796867829885, 12.49967909400553

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