Properties

Label 2-394944-1.1-c1-0-177
Degree $2$
Conductor $394944$
Sign $-1$
Analytic cond. $3153.64$
Root an. cond. $56.1573$
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 + 2·5-s + 4·7-s + 9-s − 4·13-s − 2·15-s + 17-s − 8·19-s − 4·21-s − 25-s − 27-s + 10·31-s + 8·35-s − 8·37-s + 4·39-s + 10·41-s − 8·43-s + 2·45-s + 10·47-s + 9·49-s − 51-s + 12·53-s + 8·57-s + 8·59-s − 2·61-s + 4·63-s − 8·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 1.10·13-s − 0.516·15-s + 0.242·17-s − 1.83·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s + 1.79·31-s + 1.35·35-s − 1.31·37-s + 0.640·39-s + 1.56·41-s − 1.21·43-s + 0.298·45-s + 1.45·47-s + 9/7·49-s − 0.140·51-s + 1.64·53-s + 1.05·57-s + 1.04·59-s − 0.256·61-s + 0.503·63-s − 0.992·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(394944\)    =    \(2^{6} \cdot 3 \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(3153.64\)
Root analytic conductor: \(56.1573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 394944,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.60235194059869, −12.06009327722039, −11.84176302332632, −11.39522734960297, −10.73024493330498, −10.41042565199746, −10.18257117005643, −9.623004952868779, −9.005933396648362, −8.530348373561420, −8.208647160883161, −7.643577502267691, −7.056791670371974, −6.776132771698672, −6.074248077682699, −5.611715133331400, −5.368514703287414, −4.672467997200503, −4.351913418158305, −4.018719699484635, −2.902046300853925, −2.406893428179430, −1.991624449581565, −1.482220331043264, −0.8184830006236192, 0, 0.8184830006236192, 1.482220331043264, 1.991624449581565, 2.406893428179430, 2.902046300853925, 4.018719699484635, 4.351913418158305, 4.672467997200503, 5.368514703287414, 5.611715133331400, 6.074248077682699, 6.776132771698672, 7.056791670371974, 7.643577502267691, 8.208647160883161, 8.530348373561420, 9.005933396648362, 9.623004952868779, 10.18257117005643, 10.41042565199746, 10.73024493330498, 11.39522734960297, 11.84176302332632, 12.06009327722039, 12.60235194059869

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