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Properties

Label 2-379050-1.1-c1-0-170
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 + 6·11-s + 12-s − 13-s + 14-s + 16-s − 3·17-s − 18-s − 21-s − 6·22-s + 3·23-s − 24-s + 26-s + 27-s − 28-s − 3·29-s − 5·31-s − 32-s + 6·33-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 + 1.80·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.218·21-s − 1.27·22-s + 0.625·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.557·29-s − 0.898·31-s − 0.176·32-s + 1.04·33-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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−12.55975870125564, −12.29620661961323, −11.58468793919393, −11.51629676322618, −10.76349485541528, −10.43256480632025, −9.840661100085210, −9.391167691906035, −9.115346759904528, −8.656252537391360, −8.437603962827347, −7.595354608882836, −7.221231576193394, −6.736564201284952, −6.552030061527862, −5.904643898620909, −5.189332610796095, −4.780800915194148, −3.902345212307700, −3.662537309937920, −3.264783106400818, −2.379562018021842, −1.983700319981741, −1.444967214845745, −0.7898023864812071, 0, 0.7898023864812071, 1.444967214845745, 1.983700319981741, 2.379562018021842, 3.264783106400818, 3.662537309937920, 3.902345212307700, 4.780800915194148, 5.189332610796095, 5.904643898620909, 6.552030061527862, 6.736564201284952, 7.221231576193394, 7.595354608882836, 8.437603962827347, 8.656252537391360, 9.115346759904528, 9.391167691906035, 9.840661100085210, 10.43256480632025, 10.76349485541528, 11.51629676322618, 11.58468793919393, 12.29620661961323, 12.55975870125564

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