Properties

Label 2-159936-1.1-c1-0-50
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 9-s − 6·13-s − 2·15-s − 17-s + 8·23-s − 25-s + 27-s + 6·29-s − 8·31-s − 10·37-s − 6·39-s + 6·41-s + 12·43-s − 2·45-s − 51-s + 10·53-s + 8·59-s + 6·61-s + 12·65-s + 12·67-s + 8·69-s + 6·73-s − 75-s + 8·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.66·13-s − 0.516·15-s − 0.242·17-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 1.64·37-s − 0.960·39-s + 0.937·41-s + 1.82·43-s − 0.298·45-s − 0.140·51-s + 1.37·53-s + 1.04·59-s + 0.768·61-s + 1.48·65-s + 1.46·67-s + 0.963·69-s + 0.702·73-s − 0.115·75-s + 0.900·79-s + 1/9·81-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: \(0\)
Selberg data: \((2,\ 159936,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.074986690\)
\(L(\frac12)\) \(\approx\) \(2.074986690\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
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

−13.02037265754001, −12.85250248353671, −12.41256073776550, −11.84542174775340, −11.50599041853700, −10.78487153304435, −10.55529526242800, −9.804025891454910, −9.447702244294303, −8.871387265251011, −8.531015187442713, −7.899175355871409, −7.398768249125269, −7.031167571965050, −6.807613406515492, −5.737340630905546, −5.270350576437094, −4.763904587223193, −4.206065192230252, −3.690496682515542, −3.151441561935036, −2.398860640664225, −2.194739188063611, −1.067616928800154, −0.4504609824128618, 0.4504609824128618, 1.067616928800154, 2.194739188063611, 2.398860640664225, 3.151441561935036, 3.690496682515542, 4.206065192230252, 4.763904587223193, 5.270350576437094, 5.737340630905546, 6.807613406515492, 7.031167571965050, 7.398768249125269, 7.899175355871409, 8.531015187442713, 8.871387265251011, 9.447702244294303, 9.804025891454910, 10.55529526242800, 10.78487153304435, 11.50599041853700, 11.84542174775340, 12.41256073776550, 12.85250248353671, 13.02037265754001

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