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Properties

Label 2-379050-1.1-c1-0-110
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 − 3·11-s + 12-s − 4·13-s + 14-s + 16-s − 18-s − 21-s + 3·22-s − 3·23-s − 24-s + 4·26-s + 27-s − 28-s + 9·29-s + 10·31-s − 32-s − 3·33-s + 36-s + 2·37-s − 4·39-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 − 0.904·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.218·21-s + 0.639·22-s − 0.625·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.188·28-s + 1.67·29-s + 1.79·31-s − 0.176·32-s − 0.522·33-s + 1/6·36-s + 0.328·37-s − 0.640·39-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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−12.59731256719056, −12.25269154423268, −11.71606155266222, −11.50197702281782, −10.61375032493617, −10.26378627591042, −9.929710922429188, −9.747409490191800, −9.069039159662922, −8.504714973858310, −8.101796545413906, −7.937988054337778, −7.284152778017147, −6.778278406248202, −6.396252879028938, −5.939738002419843, −5.000319473769863, −4.865791521750372, −4.289665115392034, −3.410245520621333, −2.952570959304915, −2.680198388212322, −2.019271684650796, −1.462796228118438, −0.6286433379370121, 0, 0.6286433379370121, 1.462796228118438, 2.019271684650796, 2.680198388212322, 2.952570959304915, 3.410245520621333, 4.289665115392034, 4.865791521750372, 5.000319473769863, 5.939738002419843, 6.396252879028938, 6.778278406248202, 7.284152778017147, 7.937988054337778, 8.101796545413906, 8.504714973858310, 9.069039159662922, 9.747409490191800, 9.929710922429188, 10.26378627591042, 10.61375032493617, 11.50197702281782, 11.71606155266222, 12.25269154423268, 12.59731256719056

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