Properties

Label 2-119952-1.1-c1-0-102
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 − 6·11-s − 17-s + 3·19-s + 7·23-s − 4·25-s − 6·29-s + 8·31-s + 7·37-s + 9·43-s − 6·47-s − 2·53-s + 6·55-s − 11·59-s − 6·61-s − 9·67-s − 9·71-s − 12·73-s + 8·79-s + 4·83-s + 85-s − 5·89-s − 3·95-s − 4·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.80·11-s − 0.242·17-s + 0.688·19-s + 1.45·23-s − 4/5·25-s − 1.11·29-s + 1.43·31-s + 1.15·37-s + 1.37·43-s − 0.875·47-s − 0.274·53-s + 0.809·55-s − 1.43·59-s − 0.768·61-s − 1.09·67-s − 1.06·71-s − 1.40·73-s + 0.900·79-s + 0.439·83-s + 0.108·85-s − 0.529·89-s − 0.307·95-s − 0.406·97-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 \( 1 + T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.69255701909404, −13.22775444937341, −12.96293793638575, −12.45396880262638, −11.75108106340454, −11.42604477855795, −10.90377799436877, −10.44330044280368, −10.02676384447803, −9.265240705651378, −9.080343506583137, −8.218446682339739, −7.785995646248282, −7.542150594645278, −7.026058065430539, −6.122930675728388, −5.841904846759614, −5.150951654265394, −4.643644551141169, −4.269396509263675, −3.257377070123255, −2.986563930643171, −2.401790017830965, −1.579125577851934, −0.7310804438124568, 0, 0.7310804438124568, 1.579125577851934, 2.401790017830965, 2.986563930643171, 3.257377070123255, 4.269396509263675, 4.643644551141169, 5.150951654265394, 5.841904846759614, 6.122930675728388, 7.026058065430539, 7.542150594645278, 7.785995646248282, 8.218446682339739, 9.080343506583137, 9.265240705651378, 10.02676384447803, 10.44330044280368, 10.90377799436877, 11.42604477855795, 11.75108106340454, 12.45396880262638, 12.96293793638575, 13.22775444937341, 13.69255701909404

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