Properties

Label 2-159936-1.1-c1-0-122
Degree $2$
Conductor $159936$
Sign $-1$
Analytic cond. $1277.09$
Root an. cond. $35.7364$
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  + 3-s − 4·5-s + 9-s − 4·11-s − 13-s − 4·15-s + 17-s − 19-s + 2·23-s + 11·25-s + 27-s − 6·29-s − 9·31-s − 4·33-s + 11·37-s − 39-s + 10·41-s + 7·43-s − 4·45-s + 6·47-s + 51-s + 6·53-s + 16·55-s − 57-s − 8·59-s − 6·61-s + 4·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 1.03·15-s + 0.242·17-s − 0.229·19-s + 0.417·23-s + 11/5·25-s + 0.192·27-s − 1.11·29-s − 1.61·31-s − 0.696·33-s + 1.80·37-s − 0.160·39-s + 1.56·41-s + 1.06·43-s − 0.596·45-s + 0.875·47-s + 0.140·51-s + 0.824·53-s + 2.15·55-s − 0.132·57-s − 1.04·59-s − 0.768·61-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(159936\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(1277.09\)
Root analytic conductor: \(35.7364\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{159936} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 159936,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
17 \( 1 - T \)
good5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
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

−13.37296854548519, −12.93756549496944, −12.56233870525140, −12.28171450974482, −11.49639520534105, −11.13566199112188, −10.78003474492560, −10.34803006724795, −9.479322614053074, −9.185669288573607, −8.652349587730801, −8.034185758890815, −7.693483020677134, −7.304337887161577, −7.166578207565536, −6.023653059691776, −5.667666844194506, −4.921889334500774, −4.349551337832394, −4.018763005849694, −3.467400881234200, −2.699587226933139, −2.572152271692983, −1.497961849415312, −0.6534491340569341, 0, 0.6534491340569341, 1.497961849415312, 2.572152271692983, 2.699587226933139, 3.467400881234200, 4.018763005849694, 4.349551337832394, 4.921889334500774, 5.667666844194506, 6.023653059691776, 7.166578207565536, 7.304337887161577, 7.693483020677134, 8.034185758890815, 8.652349587730801, 9.185669288573607, 9.479322614053074, 10.34803006724795, 10.78003474492560, 11.13566199112188, 11.49639520534105, 12.28171450974482, 12.56233870525140, 12.93756549496944, 13.37296854548519

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