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Properties

Label 2-1680-20.7-c1-0-27
Degree $2$
Conductor $1680$
Sign $-0.633 + 0.774i$
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 + (0.426 − 2.19i)5-s + (−0.707 + 0.707i)7-s + 1.00i·9-s − 4.62i·11-s + (4.46 − 4.46i)13-s + (−1.85 + 1.25i)15-s + (3.46 + 3.46i)17-s − 0.327·19-s + 1.00·21-s + (1.24 + 1.24i)23-s + (−4.63 − 1.87i)25-s + (0.707 − 0.707i)27-s + 2.56i·29-s − 1.14i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.190 − 0.981i)5-s + (−0.267 + 0.267i)7-s + 0.333i·9-s − 1.39i·11-s + (1.23 − 1.23i)13-s + (−0.478 + 0.322i)15-s + (0.840 + 0.840i)17-s − 0.0751·19-s + 0.218·21-s + (0.259 + 0.259i)23-s + (−0.927 − 0.374i)25-s + (0.136 − 0.136i)27-s + 0.476i·29-s − 0.205i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.633 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.633 + 0.774i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.633 + 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.359079274\)
\(L(\frac12)\) \(\approx\) \(1.359079274\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.426 + 2.19i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + 4.62iT - 11T^{2} \)
13 \( 1 + (-4.46 + 4.46i)T - 13iT^{2} \)
17 \( 1 + (-3.46 - 3.46i)T + 17iT^{2} \)
19 \( 1 + 0.327T + 19T^{2} \)
23 \( 1 + (-1.24 - 1.24i)T + 23iT^{2} \)
29 \( 1 - 2.56iT - 29T^{2} \)
31 \( 1 + 1.14iT - 31T^{2} \)
37 \( 1 + (5.99 + 5.99i)T + 37iT^{2} \)
41 \( 1 - 7.43T + 41T^{2} \)
43 \( 1 + (3.16 + 3.16i)T + 43iT^{2} \)
47 \( 1 + (4.49 - 4.49i)T - 47iT^{2} \)
53 \( 1 + (-9.83 + 9.83i)T - 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 1.11T + 61T^{2} \)
67 \( 1 + (-2.03 + 2.03i)T - 67iT^{2} \)
71 \( 1 - 3.35iT - 71T^{2} \)
73 \( 1 + (2.61 - 2.61i)T - 73iT^{2} \)
79 \( 1 - 6.92T + 79T^{2} \)
83 \( 1 + (6.02 + 6.02i)T + 83iT^{2} \)
89 \( 1 - 1.06iT - 89T^{2} \)
97 \( 1 + (13.2 + 13.2i)T + 97iT^{2} \)
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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

−8.773679810257930007030848857884, −8.417998263832025707201433570836, −7.67059856364524884897144155165, −6.38182493033221780319556980224, −5.65707848788244374461226777443, −5.41552512327416070338777850172, −3.91165670615569603904642456090, −3.11356718988483646021702662939, −1.52352458343619146658917197475, −0.58893239492772617635530971175, 1.51999090328794140293384624107, 2.79182991715815344329058882372, 3.82312137166643108025926018571, 4.57414065184599406164615809479, 5.64845523825012154972766672813, 6.62191078124350653438288696699, 6.95725140739535196060966351096, 7.923768539797697279544715568757, 9.161430170841613919271005439582, 9.682939651232891967268534528916

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