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Properties

Label 2-119952-1.1-c1-0-134
Degree $2$
Conductor $119952$
Sign $-1$
Analytic cond. $957.821$
Root an. cond. $30.9486$
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  − 2·13-s − 17-s + 4·19-s + 2·23-s − 5·25-s + 6·31-s − 10·41-s − 4·43-s + 4·47-s + 2·53-s + 4·59-s − 4·67-s − 2·71-s + 14·73-s − 6·79-s + 12·83-s − 2·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.554·13-s − 0.242·17-s + 0.917·19-s + 0.417·23-s − 25-s + 1.07·31-s − 1.56·41-s − 0.609·43-s + 0.583·47-s + 0.274·53-s + 0.520·59-s − 0.488·67-s − 0.237·71-s + 1.63·73-s − 0.675·79-s + 1.31·83-s − 0.211·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119952\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(957.821\)
Root analytic conductor: \(30.9486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 119952,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
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

−13.69595141264534, −13.49884999202993, −12.87543635850019, −12.20515058617188, −11.92104955825835, −11.49710320412277, −10.94232733931544, −10.25695958943242, −9.994014970993528, −9.432894611355079, −8.989513568110409, −8.273202177039757, −7.992754551774865, −7.289191756279399, −6.918510558006377, −6.333967122836380, −5.754235535095732, −5.099615874658027, −4.832457944709316, −4.026093972683984, −3.524007120677757, −2.871858735123834, −2.295069053143534, −1.596373345071720, −0.8478750777666299, 0, 0.8478750777666299, 1.596373345071720, 2.295069053143534, 2.871858735123834, 3.524007120677757, 4.026093972683984, 4.832457944709316, 5.099615874658027, 5.754235535095732, 6.333967122836380, 6.918510558006377, 7.289191756279399, 7.992754551774865, 8.273202177039757, 8.989513568110409, 9.432894611355079, 9.994014970993528, 10.25695958943242, 10.94232733931544, 11.49710320412277, 11.92104955825835, 12.20515058617188, 12.87543635850019, 13.49884999202993, 13.69595141264534

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