Properties

Label 2-188760-1.1-c1-0-35
Degree $2$
Conductor $188760$
Sign $-1$
Analytic cond. $1507.25$
Root an. cond. $38.8233$
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 − 5-s − 2·7-s + 9-s + 13-s + 15-s + 3·17-s − 4·19-s + 2·21-s + 5·23-s + 25-s − 27-s − 9·31-s + 2·35-s + 2·37-s − 39-s + 4·43-s − 45-s + 13·47-s − 3·49-s − 3·51-s − 53-s + 4·57-s − 6·59-s + 13·61-s − 2·63-s − 65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 0.727·17-s − 0.917·19-s + 0.436·21-s + 1.04·23-s + 1/5·25-s − 0.192·27-s − 1.61·31-s + 0.338·35-s + 0.328·37-s − 0.160·39-s + 0.609·43-s − 0.149·45-s + 1.89·47-s − 3/7·49-s − 0.420·51-s − 0.137·53-s + 0.529·57-s − 0.781·59-s + 1.66·61-s − 0.251·63-s − 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(188760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1507.25\)
Root analytic conductor: \(38.8233\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{188760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 188760,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.06845852357050, −12.75898291455812, −12.62329834708202, −11.91982497891777, −11.46532781098762, −10.96720704554322, −10.62272148684109, −10.18403206123017, −9.442538595067026, −9.250593759344037, −8.590832645262512, −8.104774553848789, −7.493219236218259, −6.913725420028556, −6.804942710112782, −5.899502225302985, −5.679910113742906, −5.134811883873390, −4.318822355426101, −4.037535885658882, −3.384282606332764, −2.872656273041907, −2.166767991069865, −1.358066672386683, −0.7035506718809189, 0, 0.7035506718809189, 1.358066672386683, 2.166767991069865, 2.872656273041907, 3.384282606332764, 4.037535885658882, 4.318822355426101, 5.134811883873390, 5.679910113742906, 5.899502225302985, 6.804942710112782, 6.913725420028556, 7.493219236218259, 8.104774553848789, 8.590832645262512, 9.250593759344037, 9.442538595067026, 10.18403206123017, 10.62272148684109, 10.96720704554322, 11.46532781098762, 11.91982497891777, 12.62329834708202, 12.75898291455812, 13.06845852357050

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