Properties

Label 2-119952-1.1-c1-0-107
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  − 2·5-s + 2·13-s + 17-s − 4·19-s + 4·23-s − 25-s − 6·29-s + 4·31-s − 2·37-s − 6·41-s − 4·43-s − 6·53-s + 12·59-s + 10·61-s − 4·65-s − 4·67-s − 4·71-s + 6·73-s − 12·79-s + 4·83-s − 2·85-s + 10·89-s + 8·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.554·13-s + 0.242·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.937·41-s − 0.609·43-s − 0.824·53-s + 1.56·59-s + 1.28·61-s − 0.496·65-s − 0.488·67-s − 0.474·71-s + 0.702·73-s − 1.35·79-s + 0.439·83-s − 0.216·85-s + 1.05·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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: $\chi_{119952} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.75058634338855, −13.14275525856240, −12.98971367159261, −12.33251499332896, −11.76391888565356, −11.42052964669101, −11.06318092409365, −10.36602318164008, −10.05326928035002, −9.357462955766257, −8.765500635868954, −8.416132819656669, −7.944333191214740, −7.381687009704073, −6.864519028854854, −6.403857008012108, −5.755866952574821, −5.175031613967831, −4.624500459790864, −3.980318559406579, −3.590335121531002, −3.042298427086742, −2.222299487655922, −1.592377904135098, −0.7536721137561170, 0, 0.7536721137561170, 1.592377904135098, 2.222299487655922, 3.042298427086742, 3.590335121531002, 3.980318559406579, 4.624500459790864, 5.175031613967831, 5.755866952574821, 6.403857008012108, 6.864519028854854, 7.381687009704073, 7.944333191214740, 8.416132819656669, 8.765500635868954, 9.357462955766257, 10.05326928035002, 10.36602318164008, 11.06318092409365, 11.42052964669101, 11.76391888565356, 12.33251499332896, 12.98971367159261, 13.14275525856240, 13.75058634338855

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