Properties

Label 2-119952-1.1-c1-0-142
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 + 2·19-s − 4·25-s + 3·29-s + 5·31-s + 4·37-s − 7·41-s − 2·43-s − 7·47-s + 6·53-s − 2·55-s − 9·59-s + 6·61-s − 65-s + 16·67-s − 2·71-s + 2·73-s − 16·79-s + 83-s − 85-s − 12·89-s − 2·95-s + 18·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.603·11-s + 0.277·13-s + 0.242·17-s + 0.458·19-s − 4/5·25-s + 0.557·29-s + 0.898·31-s + 0.657·37-s − 1.09·41-s − 0.304·43-s − 1.02·47-s + 0.824·53-s − 0.269·55-s − 1.17·59-s + 0.768·61-s − 0.124·65-s + 1.95·67-s − 0.237·71-s + 0.234·73-s − 1.80·79-s + 0.109·83-s − 0.108·85-s − 1.27·89-s − 0.205·95-s + 1.82·97-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 \( 1 - T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.82408797446619, −13.33860693474031, −12.86556936411853, −12.27013390355949, −11.75053823030422, −11.54393331857244, −11.05555564072753, −10.28495527111778, −9.907272048368868, −9.552193546602535, −8.740968897689357, −8.467273818907195, −7.881011210446978, −7.447060667366889, −6.747501309070726, −6.414111265041879, −5.799186419352346, −5.178727169621259, −4.637393669879596, −4.026987151795780, −3.553690246900796, −2.986444328559581, −2.274524790023759, −1.481141510277413, −0.9098882016566036, 0, 0.9098882016566036, 1.481141510277413, 2.274524790023759, 2.986444328559581, 3.553690246900796, 4.026987151795780, 4.637393669879596, 5.178727169621259, 5.799186419352346, 6.414111265041879, 6.747501309070726, 7.447060667366889, 7.881011210446978, 8.467273818907195, 8.740968897689357, 9.552193546602535, 9.907272048368868, 10.28495527111778, 11.05555564072753, 11.54393331857244, 11.75053823030422, 12.27013390355949, 12.86556936411853, 13.33860693474031, 13.82408797446619

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