Properties

Label 2-259182-1.1-c1-0-109
Degree $2$
Conductor $259182$
Sign $-1$
Analytic cond. $2069.57$
Root an. cond. $45.4926$
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 + 4-s − 7-s + 8-s + 4·13-s − 14-s + 16-s − 17-s − 5·19-s − 6·23-s − 5·25-s + 4·26-s − 28-s − 6·29-s + 2·31-s + 32-s − 34-s + 8·37-s − 5·38-s − 12·41-s + 4·43-s − 6·46-s + 49-s − 5·50-s + 4·52-s + 6·53-s − 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 1.14·19-s − 1.25·23-s − 25-s + 0.784·26-s − 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s − 0.171·34-s + 1.31·37-s − 0.811·38-s − 1.87·41-s + 0.609·43-s − 0.884·46-s + 1/7·49-s − 0.707·50-s + 0.554·52-s + 0.824·53-s − 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(259182\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(2069.57\)
Root analytic conductor: \(45.4926\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{259182} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 259182,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.08942186524858, −12.78330507266449, −12.10944563070542, −11.66288526487605, −11.35858613152078, −10.83582810689412, −10.27378954106790, −9.972143777988532, −9.390444885544543, −8.769777344123354, −8.351210917998953, −7.918890536098875, −7.346905986173344, −6.729788719732177, −6.350344792934067, −5.825313220812839, −5.634768953589960, −4.794277984915758, −4.207803631158683, −3.865176975198723, −3.477506539155733, −2.711733350696263, −2.065261647554036, −1.750130002523960, −0.7943338847545179, 0, 0.7943338847545179, 1.750130002523960, 2.065261647554036, 2.711733350696263, 3.477506539155733, 3.865176975198723, 4.207803631158683, 4.794277984915758, 5.634768953589960, 5.825313220812839, 6.350344792934067, 6.729788719732177, 7.346905986173344, 7.918890536098875, 8.351210917998953, 8.769777344123354, 9.390444885544543, 9.972143777988532, 10.27378954106790, 10.83582810689412, 11.35858613152078, 11.66288526487605, 12.10944563070542, 12.78330507266449, 13.08942186524858

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