Properties

Label 2-1680-21.20-c1-0-19
Degree $2$
Conductor $1680$
Sign $0.586 - 0.810i$
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  + (−1.68 + 0.396i)3-s + 5-s + (2 − 1.73i)7-s + (2.68 − 1.33i)9-s + 0.792i·11-s + 5.84i·13-s + (−1.68 + 0.396i)15-s − 1.37·17-s + 3.46i·19-s + (−2.68 + 3.71i)21-s + 1.87i·23-s + 25-s + (−4 + 3.31i)27-s − 4.25i·29-s − 3.46i·31-s + ⋯
L(s)  = 1  + (−0.973 + 0.228i)3-s + 0.447·5-s + (0.755 − 0.654i)7-s + (0.895 − 0.445i)9-s + 0.238i·11-s + 1.61i·13-s + (−0.435 + 0.102i)15-s − 0.332·17-s + 0.794i·19-s + (−0.586 + 0.810i)21-s + 0.391i·23-s + 0.200·25-s + (−0.769 + 0.638i)27-s − 0.790i·29-s − 0.622i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.586 - 0.810i$
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: \((2,\ 1680,\ (\ :1/2),\ 0.586 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.378963161\)
\(L(\frac12)\) \(\approx\) \(1.378963161\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 - 0.396i)T \)
5 \( 1 - T \)
7 \( 1 + (-2 + 1.73i)T \)
good11 \( 1 - 0.792iT - 11T^{2} \)
13 \( 1 - 5.84iT - 13T^{2} \)
17 \( 1 + 1.37T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 1.87iT - 23T^{2} \)
29 \( 1 + 4.25iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 4.74T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 6.74T + 43T^{2} \)
47 \( 1 - 7.37T + 47T^{2} \)
53 \( 1 - 8.51iT - 53T^{2} \)
59 \( 1 + 2.74T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 6.74T + 67T^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 3.37T + 79T^{2} \)
83 \( 1 + 5.48T + 83T^{2} \)
89 \( 1 - 3.25T + 89T^{2} \)
97 \( 1 + 1.08iT - 97T^{2} \)
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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

−9.643538803368506647547323381244, −8.838689203430133646601579291428, −7.67930955369150265876515477058, −7.00819996519521335585037228765, −6.19580106160943488672759150192, −5.43541688624429952774164746025, −4.35352056821340304696296510715, −4.06738347388269426945894757292, −2.16337322651037326053200198931, −1.17380517002660459494909493075, 0.69300647530436120233477833381, 1.97300157660937276103089559676, 3.08276312176775623982388886195, 4.60169155985136736104765111665, 5.25909993946880658407176896038, 5.84722953507496251675542112664, 6.70384199849461612481332392893, 7.62634985092032778504849638252, 8.393520321156728289149006190428, 9.220876909162664559091098340858

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