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Properties

Label 2-1680-28.3-c1-0-26
Degree $2$
Conductor $1680$
Sign $-0.629 + 0.776i$
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 + (1.82 + 1.91i)7-s + (−0.499 − 0.866i)9-s + (−3.23 − 1.86i)11-s − 0.751i·13-s + 0.999i·15-s + (−2.94 − 1.70i)17-s + (−2.41 − 4.18i)19-s + (2.57 − 0.627i)21-s + (−4.01 + 2.31i)23-s + (0.499 − 0.866i)25-s − 0.999·27-s + 0.0618·29-s + (4.77 − 8.27i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.387 + 0.223i)5-s + (0.691 + 0.722i)7-s + (−0.166 − 0.288i)9-s + (−0.976 − 0.563i)11-s − 0.208i·13-s + 0.258i·15-s + (−0.714 − 0.412i)17-s + (−0.554 − 0.959i)19-s + (0.560 − 0.136i)21-s + (−0.837 + 0.483i)23-s + (0.0999 − 0.173i)25-s − 0.192·27-s + 0.0114·29-s + (0.857 − 1.48i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.629 + 0.776i$
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: \((2,\ 1680,\ (\ :1/2),\ -0.629 + 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9773503929\)
\(L(\frac12)\) \(\approx\) \(0.9773503929\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 - 1.91i)T \)
good11 \( 1 + (3.23 + 1.86i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.751iT - 13T^{2} \)
17 \( 1 + (2.94 + 1.70i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.41 + 4.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.01 - 2.31i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.0618T + 29T^{2} \)
31 \( 1 + (-4.77 + 8.27i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.74 + 3.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.22iT - 41T^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 + (-0.654 - 1.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.177 + 0.307i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.68 + 8.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.85 + 1.64i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.989 + 0.571i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (-3.03 - 1.75i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.66 + 0.958i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.86T + 83T^{2} \)
89 \( 1 + (16.0 - 9.27i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.45iT - 97T^{2} \)
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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

−8.834467227786889669610314769833, −8.226929408213317402856177006129, −7.65878881252722777744296128563, −6.74254244519721035182194753756, −5.81460465869199471386605032527, −5.02946505059576275531603944536, −3.97613259839910352103110258962, −2.72735477545499470775841188326, −2.12810155778569812615441305486, −0.34147919843670674875895834113, 1.55609629571434687232570922643, 2.72981396836104479204374614565, 4.02760882956911006923877735799, 4.48250727897288191683909568658, 5.33461155352386202787201025306, 6.51386448532648083867589746945, 7.41638140418737398433545461857, 8.280328770241606842077872207495, 8.514447844136021683246262266880, 9.841743014667075986287831499412

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