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Properties

Label 2-1680-35.13-c1-0-5
Degree $2$
Conductor $1680$
Sign $-0.249 - 0.968i$
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 + (−2.17 − 0.513i)5-s + (−0.659 − 2.56i)7-s − 1.00i·9-s + 1.57·11-s + (−3.14 + 3.14i)13-s + (1.90 − 1.17i)15-s + (−4.47 − 4.47i)17-s + 6.38·19-s + (2.27 + 1.34i)21-s + (−1.38 − 1.38i)23-s + (4.47 + 2.23i)25-s + (0.707 + 0.707i)27-s − 2.19i·29-s + 1.53i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.973 − 0.229i)5-s + (−0.249 − 0.968i)7-s − 0.333i·9-s + 0.475·11-s + (−0.872 + 0.872i)13-s + (0.491 − 0.303i)15-s + (−1.08 − 1.08i)17-s + 1.46·19-s + (0.497 + 0.293i)21-s + (−0.288 − 0.288i)23-s + (0.894 + 0.446i)25-s + (0.136 + 0.136i)27-s − 0.407i·29-s + 0.276i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.249 - 0.968i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.249 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5290788724\)
\(L(\frac12)\) \(\approx\) \(0.5290788724\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 + 0.513i)T \)
7 \( 1 + (0.659 + 2.56i)T \)
good11 \( 1 - 1.57T + 11T^{2} \)
13 \( 1 + (3.14 - 3.14i)T - 13iT^{2} \)
17 \( 1 + (4.47 + 4.47i)T + 17iT^{2} \)
19 \( 1 - 6.38T + 19T^{2} \)
23 \( 1 + (1.38 + 1.38i)T + 23iT^{2} \)
29 \( 1 + 2.19iT - 29T^{2} \)
31 \( 1 - 1.53iT - 31T^{2} \)
37 \( 1 + (8.06 - 8.06i)T - 37iT^{2} \)
41 \( 1 - 4.79iT - 41T^{2} \)
43 \( 1 + (0.0831 + 0.0831i)T + 43iT^{2} \)
47 \( 1 + (-3.14 - 3.14i)T + 47iT^{2} \)
53 \( 1 + (-6.30 - 6.30i)T + 53iT^{2} \)
59 \( 1 - 7.59T + 59T^{2} \)
61 \( 1 - 3.73iT - 61T^{2} \)
67 \( 1 + (-1.84 + 1.84i)T - 67iT^{2} \)
71 \( 1 + 9.63T + 71T^{2} \)
73 \( 1 + (-2.46 + 2.46i)T - 73iT^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 + (2.70 - 2.70i)T - 83iT^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + (-8.82 - 8.82i)T + 97iT^{2} \)
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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

−9.567445416729717519855912522697, −8.961472934920529092036036208164, −7.88198848651309554626472513979, −7.02028099033794071510894326793, −6.73019410466752249440420122937, −5.23169417479920458421841924897, −4.52189174198689402672902428883, −3.92243733077785165698222990429, −2.84998286659689176391646762475, −1.04754531631423432665223638878, 0.25404171884418864424217620479, 1.96698772393444704595999992175, 3.08824432615898081222118384053, 4.00946011475746626133678406191, 5.20174006948170986990479051425, 5.79673438125753364456818513215, 6.89593573484377195594571941762, 7.38862773009007196240447358553, 8.345611121788340592297686169400, 8.942235196430547259840122762003

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