Properties

Label 2-119952-1.1-c1-0-116
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 − 3·13-s − 17-s − 2·19-s + 4·23-s − 4·25-s − 3·29-s − 31-s − 4·37-s + 3·41-s + 2·43-s + 47-s − 6·53-s − 2·55-s + 11·59-s + 2·61-s − 3·65-s + 4·67-s + 10·71-s − 2·73-s + 16·79-s − 3·83-s − 85-s − 2·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.603·11-s − 0.832·13-s − 0.242·17-s − 0.458·19-s + 0.834·23-s − 4/5·25-s − 0.557·29-s − 0.179·31-s − 0.657·37-s + 0.468·41-s + 0.304·43-s + 0.145·47-s − 0.824·53-s − 0.269·55-s + 1.43·59-s + 0.256·61-s − 0.372·65-s + 0.488·67-s + 1.18·71-s − 0.234·73-s + 1.80·79-s − 0.329·83-s − 0.108·85-s − 0.205·95-s + 0.203·97-s + 0.0995·101-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 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.85759507721390, −13.19417000892757, −12.89822132255845, −12.51204829673754, −11.83260085364107, −11.42514171445999, −10.76486530485069, −10.51013776931064, −9.856050530757601, −9.425091452228524, −9.055214748171116, −8.344664338841563, −7.876628179212635, −7.400070977801787, −6.779474103324328, −6.416220523160858, −5.576604218450580, −5.326191369301919, −4.762169918240942, −4.076468796443962, −3.529324051696877, −2.744042548490532, −2.264040353772956, −1.757024994823095, −0.7857609894296016, 0, 0.7857609894296016, 1.757024994823095, 2.264040353772956, 2.744042548490532, 3.529324051696877, 4.076468796443962, 4.762169918240942, 5.326191369301919, 5.576604218450580, 6.416220523160858, 6.779474103324328, 7.400070977801787, 7.876628179212635, 8.344664338841563, 9.055214748171116, 9.425091452228524, 9.856050530757601, 10.51013776931064, 10.76486530485069, 11.42514171445999, 11.83260085364107, 12.51204829673754, 12.89822132255845, 13.19417000892757, 13.85759507721390

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