Properties

Label 2-338130-1.1-c1-0-63
Degree $2$
Conductor $338130$
Sign $-1$
Analytic cond. $2699.98$
Root an. cond. $51.9613$
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 − 5-s − 2·7-s − 8-s + 10-s + 4·11-s − 13-s + 2·14-s + 16-s + 6·19-s − 20-s − 4·22-s + 25-s + 26-s − 2·28-s − 4·29-s − 6·31-s − 32-s + 2·35-s + 2·37-s − 6·38-s + 40-s + 10·41-s + 8·43-s + 4·44-s − 3·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s + 1.37·19-s − 0.223·20-s − 0.852·22-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 0.742·29-s − 1.07·31-s − 0.176·32-s + 0.338·35-s + 0.328·37-s − 0.973·38-s + 0.158·40-s + 1.56·41-s + 1.21·43-s + 0.603·44-s − 3/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338130\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2699.98\)
Root analytic conductor: \(51.9613\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{338130} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 338130,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.64926720291049, −12.30305231627445, −11.81830314565453, −11.42511302451895, −11.02064539566734, −10.54882688732759, −9.930057297382349, −9.483844641102532, −9.154033017099615, −8.994362853490430, −8.169375751099478, −7.670626272832640, −7.257536133424514, −7.044272493846370, −6.247068271585690, −5.987290930249590, −5.439684851659765, −4.735009727490699, −4.112375999186225, −3.639442295322853, −3.221385268091578, −2.608528444376297, −1.962739462699183, −1.227143751111413, −0.7636897606242136, 0, 0.7636897606242136, 1.227143751111413, 1.962739462699183, 2.608528444376297, 3.221385268091578, 3.639442295322853, 4.112375999186225, 4.735009727490699, 5.439684851659765, 5.987290930249590, 6.247068271585690, 7.044272493846370, 7.257536133424514, 7.670626272832640, 8.169375751099478, 8.994362853490430, 9.154033017099615, 9.483844641102532, 9.930057297382349, 10.54882688732759, 11.02064539566734, 11.42511302451895, 11.81830314565453, 12.30305231627445, 12.64926720291049

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