Properties

Label 2-379050-1.1-c1-0-113
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 + 3·13-s + 14-s + 16-s + 3·17-s − 18-s + 21-s + 5·22-s − 23-s + 24-s − 3·26-s − 27-s − 28-s − 3·29-s − 10·31-s − 32-s + 5·33-s − 3·34-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 − 1.50·11-s − 0.288·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.218·21-s + 1.06·22-s − 0.208·23-s + 0.204·24-s − 0.588·26-s − 0.192·27-s − 0.188·28-s − 0.557·29-s − 1.79·31-s − 0.176·32-s + 0.870·33-s − 0.514·34-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: \(1\)
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 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−12.69761710069155, −12.23816317602019, −11.67315416333099, −11.24386374837711, −10.83537703120756, −10.42941633227307, −10.12224841784203, −9.562655784142560, −9.152790918819532, −8.619752058516981, −8.098715926103398, −7.576460882403320, −7.445737012487133, −6.711694833583122, −6.239626637062214, −5.728976491191425, −5.413277668351892, −4.931415239997504, −4.173841484839473, −3.506313782283619, −3.265755262311128, −2.344345005942894, −2.068780741134423, −1.213861778031358, −0.6341411205832640, 0, 0.6341411205832640, 1.213861778031358, 2.068780741134423, 2.344345005942894, 3.265755262311128, 3.506313782283619, 4.173841484839473, 4.931415239997504, 5.413277668351892, 5.728976491191425, 6.239626637062214, 6.711694833583122, 7.445737012487133, 7.576460882403320, 8.098715926103398, 8.619752058516981, 9.152790918819532, 9.562655784142560, 10.12224841784203, 10.42941633227307, 10.83537703120756, 11.24386374837711, 11.67315416333099, 12.23816317602019, 12.69761710069155

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