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Properties

Label 2-1680-420.59-c0-0-1
Degree $2$
Conductor $1680$
Sign $0.605 + 0.795i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + 9-s + (0.5 + 0.866i)15-s + (0.5 − 0.866i)21-s + (1.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s − 27-s − 1.73i·29-s + 0.999·35-s + 41-s + 43-s + (−0.5 − 0.866i)45-s + (−1 − 1.73i)47-s + ⋯
L(s)  = 1  − 3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + 9-s + (0.5 + 0.866i)15-s + (0.5 − 0.866i)21-s + (1.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s − 27-s − 1.73i·29-s + 0.999·35-s + 41-s + 43-s + (−0.5 − 0.866i)45-s + (−1 − 1.73i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :0),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6541606194\)
\(L(\frac12)\) \(\approx\) \(0.6541606194\)
\(L(1)\) 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 \( 1 + T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.431916652567385851969390219273, −8.712744696305593854267599156768, −7.86646117947433195761318129937, −6.89354324123793379296846659127, −6.11422177266556065312387628493, −5.31630189104552531068177477352, −4.65422124867457436887237258061, −3.68222215517710984638277689370, −2.26957020275226792631940906094, −0.69883889123326455296892213699, 1.15223263733226749000786804534, 2.93928861298142540247067160683, 3.80982004378933339392575095710, 4.68930762698578407867258951814, 5.69554059771589861537870707471, 6.61067949045465340400989856893, 7.16140188500770257983797624889, 7.67847735181982922445658308561, 9.029002630290797186923713540752, 9.884068433287419927997150848639

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