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Properties

Label 2-1680-28.3-c1-0-12
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

\(L(1)\) \(\approx\) \(2.021903034\)
\(L(\frac12)\) \(\approx\) \(2.021903034\)
\(L(\frac{3}{2})\) 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

−9.098667402356872424101089952020, −8.540686793112557149093447108252, −7.892569068575217596238127916276, −6.86067462362655410551263450169, −6.47953190853361386024485258079, −5.09970661043980946534314852222, −4.49840389424487546576164513500, −3.23369483709111308945440400831, −2.30244415036369917127176802037, −1.14547721278627993371126856742, 0.934629128484154256057823938531, 2.29298765773004070738203738232, 3.65507818899952574072384743400, 4.19144817942142452067050462789, 5.10614554641690082306666833883, 5.99372461522343520948877711686, 7.16912691586844130066370607461, 7.81254195101902438368170084023, 8.641645619180263098540614253796, 9.136732779437458162800567496170

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