Properties

Label 2-1680-21.20-c1-0-12
Degree $2$
Conductor $1680$
Sign $-0.327 - 0.944i$
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  + (1.5 + 0.866i)3-s − 5-s + (−2 − 1.73i)7-s + (1.5 + 2.59i)9-s + 5.19i·11-s − 1.73i·13-s + (−1.5 − 0.866i)15-s + 3·17-s + 3.46i·19-s + (−1.50 − 4.33i)21-s + 25-s + 5.19i·27-s − 5.19i·29-s + 10.3i·31-s + (−4.5 + 7.79i)33-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s − 0.447·5-s + (−0.755 − 0.654i)7-s + (0.5 + 0.866i)9-s + 1.56i·11-s − 0.480i·13-s + (−0.387 − 0.223i)15-s + 0.727·17-s + 0.794i·19-s + (−0.327 − 0.944i)21-s + 0.200·25-s + 0.999i·27-s − 0.964i·29-s + 1.86i·31-s + (−0.783 + 1.35i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.327 - 0.944i$
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: \((2,\ 1680,\ (\ :1/2),\ -0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.581032280\)
\(L(\frac12)\) \(\approx\) \(1.581032280\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + (2 + 1.73i)T \)
good11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 5.19iT - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 - 13T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 1.73iT - 97T^{2} \)
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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

−9.733749040158929515548400794686, −8.851558402607513733528858424549, −7.911767255622625375704060215739, −7.40190163913803069998591671638, −6.63585771449316877949278701974, −5.29591712777220062768202091441, −4.41701088323047055941457096225, −3.66216343744557861561566676306, −2.88663872646298472636726172871, −1.55202335156257597807165764069, 0.53656611710741931084683969148, 2.08476272030609904313013749633, 3.26450710803956470259550416904, 3.54415772029358230774224103448, 5.01702267698658122950743633720, 6.12237931287808580117975900809, 6.67623566099226817054563724135, 7.68414124888335670709591020634, 8.356976894471798705370994653717, 9.050455057028632390964115449926

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