Properties

Label 2-119952-1.1-c1-0-109
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 − 4·11-s − 2·13-s − 17-s − 4·19-s − 4·23-s − 25-s + 2·29-s + 2·37-s − 10·41-s + 4·43-s + 8·47-s − 6·53-s − 8·55-s + 8·59-s − 2·61-s − 4·65-s + 4·67-s + 12·71-s + 10·73-s − 8·79-s − 2·85-s + 6·89-s − 8·95-s − 6·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.20·11-s − 0.554·13-s − 0.242·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s + 0.371·29-s + 0.328·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s − 0.824·53-s − 1.07·55-s + 1.04·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s − 0.216·85-s + 0.635·89-s − 0.820·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.75402730452471, −13.39149871552510, −12.84655069361234, −12.44863857117192, −12.01885038346590, −11.24688771356191, −10.88417365473796, −10.27548312418580, −9.952518994636264, −9.611795829861672, −8.863714538722897, −8.405430128115408, −7.932261566467002, −7.390689887117682, −6.775734621118059, −6.256907400670067, −5.758215866263555, −5.261187161219608, −4.757673159571460, −4.163455451133952, −3.460125949253455, −2.715941349841548, −2.141906759778457, −1.921279916159363, −0.7775735373341982, 0, 0.7775735373341982, 1.921279916159363, 2.141906759778457, 2.715941349841548, 3.460125949253455, 4.163455451133952, 4.757673159571460, 5.261187161219608, 5.758215866263555, 6.256907400670067, 6.775734621118059, 7.390689887117682, 7.932261566467002, 8.405430128115408, 8.863714538722897, 9.611795829861672, 9.952518994636264, 10.27548312418580, 10.88417365473796, 11.24688771356191, 12.01885038346590, 12.44863857117192, 12.84655069361234, 13.39149871552510, 13.75402730452471

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