Properties

Label 2-379050-1.1-c1-0-15
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 − 8·17-s − 18-s + 21-s + 5·22-s − 3·23-s + 24-s + 3·26-s − 27-s − 28-s + 4·29-s − 31-s − 32-s + 5·33-s + 8·34-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 − 1.94·17-s − 0.235·18-s + 0.218·21-s + 1.06·22-s − 0.625·23-s + 0.204·24-s + 0.588·26-s − 0.192·27-s − 0.188·28-s + 0.742·29-s − 0.179·31-s − 0.176·32-s + 0.870·33-s + 1.37·34-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: $\chi_{379050} (1, \cdot )$
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 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.77542065706900, −11.98971623276134, −11.92094288572539, −11.14663299912693, −10.94186113894030, −10.26036853117772, −10.16472646434309, −9.690755681995706, −9.056637544436496, −8.586878936452719, −8.213062537131451, −7.701657418427450, −7.064857761974271, −6.868725418348109, −6.355923360804815, −5.788118836779539, −5.246145113825162, −4.813073454155206, −4.380181128283998, −3.662802974711336, −2.792629665658096, −2.695218013987642, −1.875321306605628, −1.486102463492938, −0.2686846046400371, 0, 0.2686846046400371, 1.486102463492938, 1.875321306605628, 2.695218013987642, 2.792629665658096, 3.662802974711336, 4.380181128283998, 4.813073454155206, 5.246145113825162, 5.788118836779539, 6.355923360804815, 6.868725418348109, 7.064857761974271, 7.701657418427450, 8.213062537131451, 8.586878936452719, 9.056637544436496, 9.690755681995706, 10.16472646434309, 10.26036853117772, 10.94186113894030, 11.14663299912693, 11.92094288572539, 11.98971623276134, 12.77542065706900

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