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Properties

Label 2-1680-20.3-c1-0-12
Degree $2$
Conductor $1680$
Sign $0.913 - 0.406i$
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  + (−0.707 + 0.707i)3-s + (−0.970 − 2.01i)5-s + (0.707 + 0.707i)7-s − 1.00i·9-s + 1.41i·11-s + (−1.41 − 1.41i)13-s + (2.11 + 0.738i)15-s + (−4.65 + 4.65i)17-s + 6.75·19-s − 1.00·21-s + (0.546 − 0.546i)23-s + (−3.11 + 3.90i)25-s + (0.707 + 0.707i)27-s + 4.23i·29-s − 3.75i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.433 − 0.900i)5-s + (0.267 + 0.267i)7-s − 0.333i·9-s + 0.426i·11-s + (−0.392 − 0.392i)13-s + (0.544 + 0.190i)15-s + (−1.12 + 1.12i)17-s + 1.54·19-s − 0.218·21-s + (0.113 − 0.113i)23-s + (−0.623 + 0.781i)25-s + (0.136 + 0.136i)27-s + 0.787i·29-s − 0.673i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.913 - 0.406i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.913 - 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.272342752\)
\(L(\frac12)\) \(\approx\) \(1.272342752\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.970 + 2.01i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + (1.41 + 1.41i)T + 13iT^{2} \)
17 \( 1 + (4.65 - 4.65i)T - 17iT^{2} \)
19 \( 1 - 6.75T + 19T^{2} \)
23 \( 1 + (-0.546 + 0.546i)T - 23iT^{2} \)
29 \( 1 - 4.23iT - 29T^{2} \)
31 \( 1 + 3.75iT - 31T^{2} \)
37 \( 1 + (-7.20 + 7.20i)T - 37iT^{2} \)
41 \( 1 - 6.29T + 41T^{2} \)
43 \( 1 + (-0.119 + 0.119i)T - 43iT^{2} \)
47 \( 1 + (2.16 + 2.16i)T + 47iT^{2} \)
53 \( 1 + (-6.39 - 6.39i)T + 53iT^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 2.48T + 61T^{2} \)
67 \( 1 + (-7.75 - 7.75i)T + 67iT^{2} \)
71 \( 1 - 6.94iT - 71T^{2} \)
73 \( 1 + (8.87 + 8.87i)T + 73iT^{2} \)
79 \( 1 - 9.45T + 79T^{2} \)
83 \( 1 + (-4.53 + 4.53i)T - 83iT^{2} \)
89 \( 1 - 15.8iT - 89T^{2} \)
97 \( 1 + (2.26 - 2.26i)T - 97iT^{2} \)
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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

−9.343291305229997528228676647255, −8.730027983834478720307282516243, −7.85074990068046566280270030192, −7.13672347814781276192326352930, −5.91755400464552683558089173457, −5.27495973241516901565001213939, −4.46575101574400577767866224685, −3.73807310951270807700239311680, −2.30406263578224682551545362142, −0.900782511018892446700200374369, 0.71247741697995320861609830806, 2.30479165143303162612853665547, 3.19616007244358144170752838202, 4.35373106112719856123812632600, 5.19871373030160603255056915040, 6.26043815965583435296110232792, 6.99199792329043461799978900299, 7.50870118935260923807565281961, 8.320512061038982239816775277155, 9.428819712606746243649277552791

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