Properties

Label 2-162288-1.1-c1-0-158
Degree $2$
Conductor $162288$
Sign $-1$
Analytic cond. $1295.87$
Root an. cond. $35.9982$
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  + 3·5-s + 4·11-s + 3·13-s − 4·17-s + 23-s + 4·25-s − 3·29-s − 6·31-s − 9·37-s + 9·41-s + 3·43-s + 7·47-s + 4·53-s + 12·55-s − 6·59-s − 10·61-s + 9·65-s − 4·67-s − 6·71-s + 8·73-s − 8·79-s − 4·83-s − 12·85-s − 14·89-s + 7·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.20·11-s + 0.832·13-s − 0.970·17-s + 0.208·23-s + 4/5·25-s − 0.557·29-s − 1.07·31-s − 1.47·37-s + 1.40·41-s + 0.457·43-s + 1.02·47-s + 0.549·53-s + 1.61·55-s − 0.781·59-s − 1.28·61-s + 1.11·65-s − 0.488·67-s − 0.712·71-s + 0.936·73-s − 0.900·79-s − 0.439·83-s − 1.30·85-s − 1.48·89-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162288\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(1295.87\)
Root analytic conductor: \(35.9982\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{162288} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 162288,\ (\ :1/2),\ -1)\)

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 \( 1 \)
23 \( 1 - T \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.56496156497440, −13.12944461338540, −12.55359209207653, −12.23973558253352, −11.45983497185818, −11.11150438145898, −10.63215538917462, −10.24923441331726, −9.444281212287832, −9.227653598142821, −8.896972295314024, −8.431877287850043, −7.541148142512070, −7.099374446938770, −6.609090124647291, −6.022072172291713, −5.805240356865835, −5.239453067529919, −4.432433806187555, −4.040509457126481, −3.431131753472653, −2.714357572333653, −2.056291758261890, −1.578311719423415, −1.077917672493038, 0, 1.077917672493038, 1.578311719423415, 2.056291758261890, 2.714357572333653, 3.431131753472653, 4.040509457126481, 4.432433806187555, 5.239453067529919, 5.805240356865835, 6.022072172291713, 6.609090124647291, 7.099374446938770, 7.541148142512070, 8.431877287850043, 8.896972295314024, 9.227653598142821, 9.444281212287832, 10.24923441331726, 10.63215538917462, 11.11150438145898, 11.45983497185818, 12.23973558253352, 12.55359209207653, 13.12944461338540, 13.56496156497440

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