Properties

Label 2-119952-1.1-c1-0-120
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 + 6·13-s + 17-s − 8·23-s − 25-s + 6·29-s − 8·31-s + 10·37-s − 6·41-s − 12·43-s + 10·53-s + 8·59-s − 6·61-s − 12·65-s − 12·67-s + 6·73-s + 8·79-s − 16·83-s − 2·85-s + 2·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 16·115-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.66·13-s + 0.242·17-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.937·41-s − 1.82·43-s + 1.37·53-s + 1.04·59-s − 0.768·61-s − 1.48·65-s − 1.46·67-s + 0.702·73-s + 0.900·79-s − 1.75·83-s − 0.216·85-s + 0.211·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.49·115-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 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.76442865680582, −13.27860669274940, −12.98513019181027, −12.12155898339269, −11.92271131062022, −11.42773948838140, −11.03149369724760, −10.35912162892314, −10.05690129506926, −9.402486679950182, −8.730863732818024, −8.308129887419286, −8.057866436484019, −7.416755338781729, −6.877695740933601, −6.205410192922492, −5.899182347817803, −5.266992520987224, −4.454659682833887, −4.059636022012423, −3.539303295212663, −3.134425473216619, −2.165041112327149, −1.576652452687475, −0.8013681837041943, 0, 0.8013681837041943, 1.576652452687475, 2.165041112327149, 3.134425473216619, 3.539303295212663, 4.059636022012423, 4.454659682833887, 5.266992520987224, 5.899182347817803, 6.205410192922492, 6.877695740933601, 7.416755338781729, 8.057866436484019, 8.308129887419286, 8.730863732818024, 9.402486679950182, 10.05690129506926, 10.35912162892314, 11.03149369724760, 11.42773948838140, 11.92271131062022, 12.12155898339269, 12.98513019181027, 13.27860669274940, 13.76442865680582

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