Properties

Label 2-1680-5.4-c1-0-33
Degree $2$
Conductor $1680$
Sign $-0.662 + 0.749i$
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  + i·3-s + (1.48 − 1.67i)5-s i·7-s − 9-s − 6.31·11-s + 6.96i·13-s + (1.67 + 1.48i)15-s − 6.57i·17-s − 3.73·19-s + 21-s − 5.73i·23-s + (−0.612 − 4.96i)25-s i·27-s + 2·29-s + 1.03·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.662 − 0.749i)5-s − 0.377i·7-s − 0.333·9-s − 1.90·11-s + 1.93i·13-s + (0.432 + 0.382i)15-s − 1.59i·17-s − 0.857·19-s + 0.218·21-s − 1.19i·23-s + (−0.122 − 0.992i)25-s − 0.192i·27-s + 0.371·29-s + 0.186·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6255701579\)
\(L(\frac12)\) \(\approx\) \(0.6255701579\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (-1.48 + 1.67i)T \)
7 \( 1 + iT \)
good11 \( 1 + 6.31T + 11T^{2} \)
13 \( 1 - 6.96iT - 13T^{2} \)
17 \( 1 + 6.57iT - 17T^{2} \)
19 \( 1 + 3.73T + 19T^{2} \)
23 \( 1 + 5.73iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 1.03T + 31T^{2} \)
37 \( 1 + 10.7iT - 37T^{2} \)
41 \( 1 + 6.96T + 41T^{2} \)
43 \( 1 - 5.92iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 1.03iT - 53T^{2} \)
59 \( 1 + 3.22T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + 4.77iT - 67T^{2} \)
71 \( 1 + 8.23T + 71T^{2} \)
73 \( 1 - 4.26iT - 73T^{2} \)
79 \( 1 + 5.92T + 79T^{2} \)
83 \( 1 + 3.22iT - 83T^{2} \)
89 \( 1 + 2.18T + 89T^{2} \)
97 \( 1 + 3.73iT - 97T^{2} \)
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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

−9.109336675129902428612616631377, −8.499293400610553827840266446100, −7.50597449354321976084908849805, −6.60905229423063489631935744433, −5.66330085344481926366313201118, −4.61897462494894681333250983222, −4.53228677761958922868746905640, −2.84820395713762471101419538693, −2.00503786719265638891576340206, −0.21429298850769099316222447182, 1.68771914612339091273721285716, 2.72872323708916814607659445729, 3.30987063563116237682606979886, 5.04062691819215566949247818014, 5.72757461384280251470965652158, 6.25230382264972743836373607240, 7.38392543707331839432484556545, 8.056093589845106043913664984747, 8.538604960543851814293977950064, 9.938354255234116218907455598776

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