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Properties

Label 2-379050-1.1-c1-0-138
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 + 4·13-s − 14-s + 16-s + 6·17-s + 18-s + 21-s + 4·22-s + 8·23-s − 24-s + 4·26-s − 27-s − 28-s − 2·29-s + 8·31-s + 32-s − 4·33-s + 6·34-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 + 1.20·11-s − 0.288·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.218·21-s + 0.852·22-s + 1.66·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.696·33-s + 1.02·34-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

\(L(1)\) \(\approx\) \(6.454522599\)
\(L(\frac12)\) \(\approx\) \(6.454522599\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.60020636780164, −12.01815471123732, −11.49851882874193, −11.26581387055315, −10.98914433379358, −10.16888166222366, −9.816002800849395, −9.540934102118870, −8.678508852535297, −8.537206933513281, −7.722821039901054, −7.348551810161320, −6.714704690599477, −6.378817571531947, −6.060362714408407, −5.502172199578265, −5.022820636780861, −4.494643425936066, −3.913344541590135, −3.541296958050794, −3.036731751066115, −2.475701631037903, −1.500949115382517, −1.111646990142345, −0.7030738079922616, 0.7030738079922616, 1.111646990142345, 1.500949115382517, 2.475701631037903, 3.036731751066115, 3.541296958050794, 3.913344541590135, 4.494643425936066, 5.022820636780861, 5.502172199578265, 6.060362714408407, 6.378817571531947, 6.714704690599477, 7.348551810161320, 7.722821039901054, 8.537206933513281, 8.678508852535297, 9.540934102118870, 9.816002800849395, 10.16888166222366, 10.98914433379358, 11.26581387055315, 11.49851882874193, 12.01815471123732, 12.60020636780164

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