Properties

Degree $2$
Conductor $1680$
Sign $0.975 - 0.218i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 + 0.618i)3-s − 5-s + (−2.61 − 0.381i)7-s + (2.23 − 2.00i)9-s + 4.47i·11-s − 1.23i·13-s + (1.61 − 0.618i)15-s − 5.23·17-s − 8.47i·19-s + (4.47 − 1.00i)21-s − 4i·23-s + 25-s + (−2.38 + 4.61i)27-s + 7.70i·29-s + 2.76i·31-s + ⋯
L(s)  = 1  + (−0.934 + 0.356i)3-s − 0.447·5-s + (−0.989 − 0.144i)7-s + (0.745 − 0.666i)9-s + 1.34i·11-s − 0.342i·13-s + (0.417 − 0.159i)15-s − 1.26·17-s − 1.94i·19-s + (0.975 − 0.218i)21-s − 0.834i·23-s + 0.200·25-s + (−0.458 + 0.888i)27-s + 1.43i·29-s + 0.496i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.975 - 0.218i$
Motivic weight: \(1\)
Character: $\chi_{1680} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.975 - 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7256863884\)
\(L(\frac12)\) \(\approx\) \(0.7256863884\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 - 0.618i)T \)
5 \( 1 + T \)
7 \( 1 + (2.61 + 0.381i)T \)
good11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + 1.23iT - 13T^{2} \)
17 \( 1 + 5.23T + 17T^{2} \)
19 \( 1 + 8.47iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 7.70iT - 29T^{2} \)
31 \( 1 - 2.76iT - 31T^{2} \)
37 \( 1 - 0.763T + 37T^{2} \)
41 \( 1 - 2.47T + 41T^{2} \)
43 \( 1 - 4.94T + 43T^{2} \)
47 \( 1 + 6.47T + 47T^{2} \)
53 \( 1 + 0.472iT - 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 - 7.23iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 7.23iT - 71T^{2} \)
73 \( 1 + 11.2iT - 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 - 5.52T + 89T^{2} \)
97 \( 1 - 0.763iT - 97T^{2} \)
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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

−9.311783856735645338077831975628, −8.954156046806161441729402664219, −7.48359838593633285781537883827, −6.78604218205584190752907784851, −6.43840980167514747513930066702, −4.99654695194061838572790554390, −4.62978828902664395454577419583, −3.58697614332718477325193177654, −2.39748030781449297594046047643, −0.58678745570981596150205855784, 0.60884747366574297473818730934, 2.14114945594425393114103069121, 3.52823915824955971478033856130, 4.22393615570992350387789236651, 5.54702520194793154193023487394, 6.10654048442219512628034356132, 6.69909144246749563671590176725, 7.73850230072972931309969015451, 8.380194129413061115993578415359, 9.462539222980154079589368042974

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