Properties

Label 2-119952-1.1-c1-0-117
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  + 5-s − 2·11-s − 13-s + 17-s − 8·19-s − 4·25-s + 5·29-s + 3·31-s − 7·41-s + 4·43-s − 9·47-s − 2·55-s + 3·59-s − 4·61-s − 65-s + 8·67-s + 6·73-s − 8·79-s + 9·83-s + 85-s − 2·89-s − 8·95-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.603·11-s − 0.277·13-s + 0.242·17-s − 1.83·19-s − 4/5·25-s + 0.928·29-s + 0.538·31-s − 1.09·41-s + 0.609·43-s − 1.31·47-s − 0.269·55-s + 0.390·59-s − 0.512·61-s − 0.124·65-s + 0.977·67-s + 0.702·73-s − 0.900·79-s + 0.987·83-s + 0.108·85-s − 0.211·89-s − 0.820·95-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−13.69036240084726, −13.38378449462258, −12.77628874272344, −12.52969051915153, −11.85623639557020, −11.43604560240764, −10.78205379201427, −10.37837952063040, −9.954517676325810, −9.571261329703721, −8.763810355490606, −8.453568990491676, −7.954829571068404, −7.409728220738087, −6.705080461233493, −6.304741948088964, −5.885257780058624, −5.109797687687041, −4.762580328266321, −4.143333482882106, −3.475116626065172, −2.823730679840188, −2.160953188944057, −1.807871927313173, −0.7862263224898604, 0, 0.7862263224898604, 1.807871927313173, 2.160953188944057, 2.823730679840188, 3.475116626065172, 4.143333482882106, 4.762580328266321, 5.109797687687041, 5.885257780058624, 6.304741948088964, 6.705080461233493, 7.409728220738087, 7.954829571068404, 8.453568990491676, 8.763810355490606, 9.571261329703721, 9.954517676325810, 10.37837952063040, 10.78205379201427, 11.43604560240764, 11.85623639557020, 12.52969051915153, 12.77628874272344, 13.38378449462258, 13.69036240084726

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