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Properties

Label 2-379050-1.1-c1-0-108
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 + 4·11-s + 12-s − 3·13-s + 14-s + 16-s + 2·17-s + 18-s + 21-s + 4·22-s − 23-s + 24-s − 3·26-s + 27-s + 28-s − 6·31-s + 32-s + 4·33-s + 2·34-s + 36-s − 4·37-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 + 1.20·11-s + 0.288·12-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.218·21-s + 0.852·22-s − 0.208·23-s + 0.204·24-s − 0.588·26-s + 0.192·27-s + 0.188·28-s − 1.07·31-s + 0.176·32-s + 0.696·33-s + 0.342·34-s + 1/6·36-s − 0.657·37-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

\(L(1)\) \(\approx\) \(7.746384249\)
\(L(\frac12)\) \(\approx\) \(7.746384249\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 17 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.53769462896893, −12.02758630240616, −11.70263188420215, −11.27944103600076, −10.73539991600705, −10.19225385795504, −9.759297997648477, −9.368131284062891, −8.703916918035185, −8.529466900587973, −7.772591412595940, −7.293185738362210, −7.090916139559214, −6.515851078257528, −5.841374543800744, −5.496995641964534, −4.966414455515333, −4.264462976343170, −4.065792013767127, −3.473252879555155, −2.997194953072140, −2.202998653386445, −2.004807371685954, −1.216610484038069, −0.6163207466636826, 0.6163207466636826, 1.216610484038069, 2.004807371685954, 2.202998653386445, 2.997194953072140, 3.473252879555155, 4.065792013767127, 4.264462976343170, 4.966414455515333, 5.496995641964534, 5.841374543800744, 6.515851078257528, 7.090916139559214, 7.293185738362210, 7.772591412595940, 8.529466900587973, 8.703916918035185, 9.368131284062891, 9.759297997648477, 10.19225385795504, 10.73539991600705, 11.27944103600076, 11.70263188420215, 12.02758630240616, 12.53769462896893

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