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Properties

Label 2-379050-1.1-c1-0-136
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 + 14-s + 16-s + 6·17-s − 18-s + 21-s + 5·22-s + 5·23-s + 24-s − 27-s − 28-s − 3·29-s − 4·31-s − 32-s + 5·33-s − 6·34-s + 36-s + 6·37-s − 3·41-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 − 1.50·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.218·21-s + 1.06·22-s + 1.04·23-s + 0.204·24-s − 0.192·27-s − 0.188·28-s − 0.557·29-s − 0.718·31-s − 0.176·32-s + 0.870·33-s − 1.02·34-s + 1/6·36-s + 0.986·37-s − 0.468·41-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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
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

−12.62754204862591, −12.34649533491052, −11.71271717575418, −11.19676881163076, −10.82231386737169, −10.49012365823887, −10.01781607985206, −9.633725854118880, −9.123708391825285, −8.694872186292201, −8.019275638245719, −7.632495539798926, −7.285584447751404, −6.956829790003529, −6.021465126067700, −5.820613808708956, −5.448290170893980, −4.839107778877902, −4.299389030361107, −3.520859788382162, −3.074436250953111, −2.566403294327162, −1.973651291179590, −1.156202352727014, −0.6975381099256136, 0, 0.6975381099256136, 1.156202352727014, 1.973651291179590, 2.566403294327162, 3.074436250953111, 3.520859788382162, 4.299389030361107, 4.839107778877902, 5.448290170893980, 5.820613808708956, 6.021465126067700, 6.956829790003529, 7.285584447751404, 7.632495539798926, 8.019275638245719, 8.694872186292201, 9.123708391825285, 9.633725854118880, 10.01781607985206, 10.49012365823887, 10.82231386737169, 11.19676881163076, 11.71271717575418, 12.34649533491052, 12.62754204862591

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