Properties

Label 2-301665-1.1-c1-0-41
Degree $2$
Conductor $301665$
Sign $-1$
Analytic cond. $2408.80$
Root an. cond. $49.0796$
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 + 5-s + 6-s − 7-s − 3·8-s + 9-s + 10-s − 12-s − 14-s + 15-s − 16-s − 17-s + 18-s − 4·19-s − 20-s − 21-s − 3·24-s + 25-s + 27-s + 28-s − 2·29-s + 30-s + 5·32-s − 34-s − 35-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s − 0.612·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s − 0.371·29-s + 0.182·30-s + 0.883·32-s − 0.171·34-s − 0.169·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(301665\)    =    \(3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(2408.80\)
Root analytic conductor: \(49.0796\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 301665,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
17 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.04831538694839, −12.58964276330028, −12.28518236744292, −11.64385715360382, −11.05885514794304, −10.63760549181492, −10.08717221538863, −9.566383442231204, −9.176714796047980, −8.986365536577751, −8.240279045854593, −7.953822115629301, −7.299020735706699, −6.643094898886474, −6.336471368691943, −5.657644896408693, −5.490892914989544, −4.618792282090078, −4.187906772848646, −4.026899689318227, −3.110244345871055, −2.827385201178811, −2.273518298844255, −1.562292962542364, −0.7733069414217003, 0, 0.7733069414217003, 1.562292962542364, 2.273518298844255, 2.827385201178811, 3.110244345871055, 4.026899689318227, 4.187906772848646, 4.618792282090078, 5.490892914989544, 5.657644896408693, 6.336471368691943, 6.643094898886474, 7.299020735706699, 7.953822115629301, 8.240279045854593, 8.986365536577751, 9.176714796047980, 9.566383442231204, 10.08717221538863, 10.63760549181492, 11.05885514794304, 11.64385715360382, 12.28518236744292, 12.58964276330028, 13.04831538694839

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