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Properties

Label 2-1680-7.2-c1-0-11
Degree $2$
Conductor $1680$
Sign $0.922 - 0.386i$
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 + (0.866 + 2.5i)7-s + (−0.499 + 0.866i)9-s + (−1.36 − 2.36i)11-s + 5.73·13-s + 0.999·15-s + (−3.36 − 5.83i)17-s + (−1.23 + 2.13i)19-s + (1.73 − 2i)21-s + (−0.633 + 1.09i)23-s + (−0.499 − 0.866i)25-s + 0.999·27-s + 6.19·29-s + (3.23 + 5.59i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.223 + 0.387i)5-s + (0.327 + 0.944i)7-s + (−0.166 + 0.288i)9-s + (−0.411 − 0.713i)11-s + 1.58·13-s + 0.258·15-s + (−0.816 − 1.41i)17-s + (−0.282 + 0.489i)19-s + (0.377 − 0.436i)21-s + (−0.132 + 0.228i)23-s + (−0.0999 − 0.173i)25-s + 0.192·27-s + 1.15·29-s + (0.580 + 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.922 - 0.386i$
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: \((2,\ 1680,\ (\ :1/2),\ 0.922 - 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.494718071\)
\(L(\frac12)\) \(\approx\) \(1.494718071\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 2.5i)T \)
good11 \( 1 + (1.36 + 2.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.73T + 13T^{2} \)
17 \( 1 + (3.36 + 5.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.23 - 2.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.633 - 1.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 + (-3.23 - 5.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.59 - 6.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.73T + 41T^{2} \)
43 \( 1 - 7.19T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.19 - 7.26i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.09 - 8.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.33 - 2.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.19T + 71T^{2} \)
73 \( 1 + (-2.33 - 4.03i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.69 + 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.12T + 83T^{2} \)
89 \( 1 + (-4.56 + 7.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.07T + 97T^{2} \)
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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

−9.120514314705957897948787763036, −8.569298056138184235674761007375, −7.937393040472896210701724706416, −6.87992626747246520371404251776, −6.18983717311986688547857813221, −5.50338409206048465541359940455, −4.51642273204914335442525004002, −3.23043317096804768331047393302, −2.42795099006746664575324357886, −1.03823794449243299381288736200, 0.75560813384656464123024486625, 2.10609265633179177477045624820, 3.76162379294957265087118159629, 4.18536572105183433001817553627, 5.02445757658769768998337860414, 6.13897844993060933111956763557, 6.76575841889937934428922247720, 7.932653501489679921970989883502, 8.432358219623757758266533586081, 9.294582866218538995942702846794

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