Properties

Label 2-1680-21.20-c1-0-27
Degree $2$
Conductor $1680$
Sign $0.654 + 0.755i$
Analytic cond. $13.4148$
Root an. cond. $3.66263$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s − 5-s + (−2 + 1.73i)7-s − 2.99·9-s + 3.46i·11-s + 1.73i·15-s + 6·17-s − 3.46i·19-s + (2.99 + 3.46i)21-s − 3.46i·23-s + 25-s + 5.19i·27-s − 6.92i·29-s + 3.46i·31-s + 5.99·33-s + ⋯
L(s)  = 1  − 0.999i·3-s − 0.447·5-s + (−0.755 + 0.654i)7-s − 0.999·9-s + 1.04i·11-s + 0.447i·15-s + 1.45·17-s − 0.794i·19-s + (0.654 + 0.755i)21-s − 0.722i·23-s + 0.200·25-s + 0.999i·27-s − 1.28i·29-s + 0.622i·31-s + 1.04·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.654 + 0.755i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.304568991\)
\(L(\frac12)\) \(\approx\) \(1.304568991\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
5 \( 1 + T \)
7 \( 1 + (2 - 1.73i)T \)
good11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−9.163571076100335377451718206971, −8.368801190515624475181747662042, −7.51555577643464559491547712804, −6.99531770856166885083132845989, −6.08534575734793081410296563567, −5.36558660343883412833402128804, −4.17110607161746032592721463974, −2.97299256042288683656230736871, −2.23283319459661454064838374185, −0.72206754198545490062980935577, 0.858340131635840093266484899711, 2.93633057642367625802991181019, 3.59652540692593678369843025719, 4.20038993135112679129094501772, 5.54839783082297697438556053599, 5.90498972589107391443652397480, 7.21463024851942921108605313917, 7.902039734334558043555125042796, 8.825883601165449380528531189619, 9.480696064539601407731812953031

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