Properties

Label 2-119952-1.1-c1-0-121
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 − 6·11-s − 4·13-s + 17-s + 2·19-s + 6·23-s − 25-s − 8·29-s + 4·31-s + 2·37-s + 10·41-s − 4·43-s − 12·47-s − 2·53-s − 12·55-s + 12·59-s + 14·61-s − 8·65-s − 4·67-s − 14·71-s − 6·73-s − 8·79-s − 4·83-s + 2·85-s − 2·89-s + 4·95-s + 14·97-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.80·11-s − 1.10·13-s + 0.242·17-s + 0.458·19-s + 1.25·23-s − 1/5·25-s − 1.48·29-s + 0.718·31-s + 0.328·37-s + 1.56·41-s − 0.609·43-s − 1.75·47-s − 0.274·53-s − 1.61·55-s + 1.56·59-s + 1.79·61-s − 0.992·65-s − 0.488·67-s − 1.66·71-s − 0.702·73-s − 0.900·79-s − 0.439·83-s + 0.216·85-s − 0.211·89-s + 0.410·95-s + 1.42·97-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.65245498719631, −13.17366483322607, −12.96048219006437, −12.67196443522077, −11.66280569082390, −11.52497772139257, −10.81597500290696, −10.30306057519553, −9.821135637950346, −9.677024401890959, −8.981541789993331, −8.352625969868488, −7.839370393446717, −7.312846099866024, −7.040074043444009, −6.161371309458205, −5.623674643305611, −5.308247902641039, −4.832946252944214, −4.231499663241081, −3.229338449740438, −2.859246609010784, −2.279766330965124, −1.761818864794364, −0.8062903177179745, 0, 0.8062903177179745, 1.761818864794364, 2.279766330965124, 2.859246609010784, 3.229338449740438, 4.231499663241081, 4.832946252944214, 5.308247902641039, 5.623674643305611, 6.161371309458205, 7.040074043444009, 7.312846099866024, 7.839370393446717, 8.352625969868488, 8.981541789993331, 9.677024401890959, 9.821135637950346, 10.30306057519553, 10.81597500290696, 11.52497772139257, 11.66280569082390, 12.67196443522077, 12.96048219006437, 13.17366483322607, 13.65245498719631

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