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Properties

Label 2-1680-7.2-c1-0-3
Degree $2$
Conductor $1680$
Sign $-0.966 - 0.254i$
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 + (−0.5 + 0.866i)5-s + (0.835 − 2.51i)7-s + (−0.499 + 0.866i)9-s + (0.335 + 0.580i)11-s − 3.51·13-s − 0.999·15-s + (−3.25 − 5.64i)17-s + (−4.01 + 6.95i)19-s + (2.59 − 0.531i)21-s + (−2.84 + 4.93i)23-s + (−0.499 − 0.866i)25-s − 0.999·27-s − 5.85·29-s + (3.09 + 5.35i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.223 + 0.387i)5-s + (0.315 − 0.948i)7-s + (−0.166 + 0.288i)9-s + (0.101 + 0.175i)11-s − 0.974·13-s − 0.258·15-s + (−0.789 − 1.36i)17-s + (−0.920 + 1.59i)19-s + (0.565 − 0.116i)21-s + (−0.593 + 1.02i)23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s − 1.08·29-s + (0.555 + 0.961i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.966 - 0.254i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.966 - 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5757858059\)
\(L(\frac12)\) \(\approx\) \(0.5757858059\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.835 + 2.51i)T \)
good11 \( 1 + (-0.335 - 0.580i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.51T + 13T^{2} \)
17 \( 1 + (3.25 + 5.64i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.01 - 6.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.84 - 4.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.85T + 29T^{2} \)
31 \( 1 + (-3.09 - 5.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.42 - 9.39i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.35T + 41T^{2} \)
43 \( 1 - 1.67T + 43T^{2} \)
47 \( 1 + (-0.591 + 1.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.10 + 10.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.92 - 8.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.51 - 7.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.67 - 2.90i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + (-4.34 - 7.53i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.42 + 7.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.17T + 83T^{2} \)
89 \( 1 + (1.74 - 3.01i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.02T + 97T^{2} \)
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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

−9.979883460687140658577156119127, −8.973604849350607007984607559527, −8.031858287264689987671538246295, −7.39308222040152231547583527632, −6.72433408708885272786599088171, −5.52268107647787413901198394638, −4.57239084933477740783611952947, −3.93825892772275191399496670085, −2.93078536881886041190941642574, −1.73227612837251201686535677776, 0.19287980011786603643037384363, 2.00709697825968448141800287064, 2.53866803412317864994012970757, 4.03700423633928878375821384486, 4.77696245201878202798595363753, 5.86899485935718269980433953131, 6.51749969920056766531843741972, 7.55212099608626461782757751427, 8.241236427967165667017668560502, 8.977065464202611212167927171725

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