Properties

Label 2-259182-1.1-c1-0-125
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 + 2·5-s + 7-s − 8-s − 2·10-s + 4·13-s − 14-s + 16-s + 17-s − 6·19-s + 2·20-s − 25-s − 4·26-s + 28-s + 6·29-s − 8·31-s − 32-s − 34-s + 2·35-s − 2·37-s + 6·38-s − 2·40-s − 2·41-s + 10·43-s − 2·47-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s − 0.353·8-s − 0.632·10-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.37·19-s + 0.447·20-s − 1/5·25-s − 0.784·26-s + 0.188·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.171·34-s + 0.338·35-s − 0.328·37-s + 0.973·38-s − 0.316·40-s − 0.312·41-s + 1.52·43-s − 0.291·47-s + 1/7·49-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.09227003901023, −12.47281401839476, −12.20856550580065, −11.53743106537742, −10.93287835809547, −10.77747223045649, −10.31664320338568, −9.843450261926017, −9.146296332726919, −8.996312835500581, −8.490983642658179, −7.944004391536231, −7.566625495977310, −6.849061701724946, −6.441949514262049, −5.878818318318088, −5.738807520547799, −4.883410681675846, −4.416582217925261, −3.694830272192651, −3.245191956081019, −2.386231844157046, −2.027779263524429, −1.473245086590610, −0.8796550590925364, 0, 0.8796550590925364, 1.473245086590610, 2.027779263524429, 2.386231844157046, 3.245191956081019, 3.694830272192651, 4.416582217925261, 4.883410681675846, 5.738807520547799, 5.878818318318088, 6.441949514262049, 6.849061701724946, 7.566625495977310, 7.944004391536231, 8.490983642658179, 8.996312835500581, 9.146296332726919, 9.843450261926017, 10.31664320338568, 10.77747223045649, 10.93287835809547, 11.53743106537742, 12.20856550580065, 12.47281401839476, 13.09227003901023

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