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Properties

Label 2-203280-1.1-c1-0-184
Degree $2$
Conductor $203280$
Sign $-1$
Analytic cond. $1623.19$
Root an. cond. $40.2889$
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 + 7-s + 9-s + 2·13-s + 15-s − 2·17-s + 4·19-s + 21-s + 25-s + 27-s + 2·29-s + 8·31-s + 35-s − 10·37-s + 2·39-s + 6·41-s − 4·43-s + 45-s + 49-s − 2·51-s − 2·53-s + 4·57-s + 12·59-s − 14·61-s + 63-s + 2·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.169·35-s − 1.64·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.280·51-s − 0.274·53-s + 0.529·57-s + 1.56·59-s − 1.79·61-s + 0.125·63-s + 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(203280\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1623.19\)
Root analytic conductor: \(40.2889\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 203280,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.48785704371742, −12.90047496891836, −12.30807577466964, −11.96108766117937, −11.37569458413131, −10.93582063760054, −10.40773979887711, −9.901381892748222, −9.619199929733627, −8.853327822049261, −8.603422085608144, −8.231893904105859, −7.464288731864193, −7.195749475148242, −6.540508417161907, −6.061733353131487, −5.529222362034681, −4.905234364711412, −4.484799652318305, −3.882007947044729, −3.212434922724454, −2.793766059641129, −2.171553378987851, −1.438940776817857, −1.094934370218718, 0, 1.094934370218718, 1.438940776817857, 2.171553378987851, 2.793766059641129, 3.212434922724454, 3.882007947044729, 4.484799652318305, 4.905234364711412, 5.529222362034681, 6.061733353131487, 6.540508417161907, 7.195749475148242, 7.464288731864193, 8.231893904105859, 8.603422085608144, 8.853327822049261, 9.619199929733627, 9.901381892748222, 10.40773979887711, 10.93582063760054, 11.37569458413131, 11.96108766117937, 12.30807577466964, 12.90047496891836, 13.48785704371742

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