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Properties

Label 2-1680-28.3-c1-0-8
Degree $2$
Conductor $1680$
Sign $0.994 - 0.106i$
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  + (−0.5 + 0.866i)3-s + (−0.866 + 0.5i)5-s + (−2.26 − 1.36i)7-s + (−0.499 − 0.866i)9-s + (−1.87 − 1.08i)11-s + 0.624i·13-s − 0.999i·15-s + (−0.335 − 0.193i)17-s + (1.76 + 3.04i)19-s + (2.31 − 1.27i)21-s + (−3.87 + 2.23i)23-s + (0.499 − 0.866i)25-s + 0.999·27-s + 9.47·29-s + (4.73 − 8.19i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.387 + 0.223i)5-s + (−0.855 − 0.517i)7-s + (−0.166 − 0.288i)9-s + (−0.565 − 0.326i)11-s + 0.173i·13-s − 0.258i·15-s + (−0.0812 − 0.0469i)17-s + (0.403 + 0.699i)19-s + (0.505 − 0.278i)21-s + (−0.807 + 0.466i)23-s + (0.0999 − 0.173i)25-s + 0.192·27-s + 1.75·29-s + (0.850 − 1.47i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.994 - 0.106i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.994 - 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.064127767\)
\(L(\frac12)\) \(\approx\) \(1.064127767\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.26 + 1.36i)T \)
good11 \( 1 + (1.87 + 1.08i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.624iT - 13T^{2} \)
17 \( 1 + (0.335 + 0.193i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.76 - 3.04i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.87 - 2.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.47T + 29T^{2} \)
31 \( 1 + (-4.73 + 8.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.85 - 3.21i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.80iT - 41T^{2} \)
43 \( 1 + 8.75iT - 43T^{2} \)
47 \( 1 + (0.432 + 0.749i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.63 + 11.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.30 - 3.99i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.08 + 2.35i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.20 - 4.15i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.64iT - 71T^{2} \)
73 \( 1 + (-14.2 - 8.22i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.03 - 0.598i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.472T + 83T^{2} \)
89 \( 1 + (-5.00 + 2.88i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.92iT - 97T^{2} \)
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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

−9.748971343381724523554740406545, −8.468097274802665377287401413262, −7.87427046785061828609027011426, −6.84780402357596438960574153592, −6.19029686319004576376760807505, −5.28495747016565682489495995206, −4.21276856395656883179043848058, −3.54922352383076601555852836072, −2.54765206596650236483793637843, −0.65407325223490777678022946389, 0.76545234922741970307196669622, 2.37762671574573488963328901792, 3.16765083021029424803842398724, 4.48627372758254576003666310790, 5.26375342858558156520747721911, 6.26002833099340604565860949742, 6.83027349563194932349892180720, 7.78230606651331174334357601908, 8.475331224858766354726974023071, 9.275889776703549864747025073461

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