Properties

Label 2-407e2-1.1-c1-0-0
Degree $2$
Conductor $165649$
Sign $1$
Analytic cond. $1322.71$
Root an. cond. $36.3691$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s − 4-s − 5-s − 2·6-s − 2·7-s + 3·8-s + 9-s + 10-s − 2·12-s − 13-s + 2·14-s − 2·15-s − 16-s + 5·17-s − 18-s − 6·19-s + 20-s − 4·21-s − 2·23-s + 6·24-s − 4·25-s + 26-s − 4·27-s + 2·28-s − 9·29-s + 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s − 0.277·13-s + 0.534·14-s − 0.516·15-s − 1/4·16-s + 1.21·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.872·21-s − 0.417·23-s + 1.22·24-s − 4/5·25-s + 0.196·26-s − 0.769·27-s + 0.377·28-s − 1.67·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165649\)    =    \(11^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(1322.71\)
Root analytic conductor: \(36.3691\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{165649} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 165649,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2739145104\)
\(L(\frac12)\) \(\approx\) \(0.2739145104\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−13.34754515671417, −12.86743754874536, −12.37472104738081, −11.91298382853245, −11.23887660485853, −10.67919482901266, −10.16791418030559, −9.758596244767909, −9.365297283084900, −8.963938510952463, −8.439293544863213, −8.011415233275497, −7.644227004309562, −7.288499416301034, −6.485166516836946, −5.940787603288865, −5.336772590487754, −4.665042034176910, −4.007212962699633, −3.563504118218012, −3.326798994843827, −2.283139276800897, −2.024899761243099, −1.134625262378888, −0.1686940904215713, 0.1686940904215713, 1.134625262378888, 2.024899761243099, 2.283139276800897, 3.326798994843827, 3.563504118218012, 4.007212962699633, 4.665042034176910, 5.336772590487754, 5.940787603288865, 6.485166516836946, 7.288499416301034, 7.644227004309562, 8.011415233275497, 8.439293544863213, 8.963938510952463, 9.365297283084900, 9.758596244767909, 10.16791418030559, 10.67919482901266, 11.23887660485853, 11.91298382853245, 12.37472104738081, 12.86743754874536, 13.34754515671417

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