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Properties

Label 2-1680-105.104-c1-0-14
Degree $2$
Conductor $1680$
Sign $-0.993 - 0.114i$
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  + (0.325 + 1.70i)3-s + (1.34 + 1.78i)5-s + (−1.53 − 2.15i)7-s + (−2.78 + 1.10i)9-s + 5.00i·11-s + 1.62·13-s + (−2.60 + 2.86i)15-s + 4.73i·17-s − 2.13i·19-s + (3.15 − 3.31i)21-s − 9.46·23-s + (−1.38 + 4.80i)25-s + (−2.79 − 4.38i)27-s − 5.96i·29-s − 4.38i·31-s + ⋯
L(s)  = 1  + (0.187 + 0.982i)3-s + (0.601 + 0.798i)5-s + (−0.582 − 0.813i)7-s + (−0.929 + 0.369i)9-s + 1.51i·11-s + 0.451·13-s + (−0.671 + 0.740i)15-s + 1.14i·17-s − 0.490i·19-s + (0.689 − 0.724i)21-s − 1.97·23-s + (−0.276 + 0.961i)25-s + (−0.537 − 0.843i)27-s − 1.10i·29-s − 0.787i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.993 - 0.114i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.993 - 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.220135815\)
\(L(\frac12)\) \(\approx\) \(1.220135815\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.325 - 1.70i)T \)
5 \( 1 + (-1.34 - 1.78i)T \)
7 \( 1 + (1.53 + 2.15i)T \)
good11 \( 1 - 5.00iT - 11T^{2} \)
13 \( 1 - 1.62T + 13T^{2} \)
17 \( 1 - 4.73iT - 17T^{2} \)
19 \( 1 + 2.13iT - 19T^{2} \)
23 \( 1 + 9.46T + 23T^{2} \)
29 \( 1 + 5.96iT - 29T^{2} \)
31 \( 1 + 4.38iT - 31T^{2} \)
37 \( 1 - 3.75iT - 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 - 4.74iT - 43T^{2} \)
47 \( 1 - 5.64iT - 47T^{2} \)
53 \( 1 + 4.69T + 53T^{2} \)
59 \( 1 + 8.66T + 59T^{2} \)
61 \( 1 + 2.09iT - 61T^{2} \)
67 \( 1 - 6.20iT - 67T^{2} \)
71 \( 1 - 8.65iT - 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 + 1.06T + 79T^{2} \)
83 \( 1 + 5.96iT - 83T^{2} \)
89 \( 1 - 0.187T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−9.815043745058959269532960222930, −9.358695205713015621350480580990, −8.064612615597783781997798062578, −7.44173531206919675438779266694, −6.27378630325128896229610168079, −5.93301818449409088525451098671, −4.39550525451914533622358632219, −4.06809264408152040526843756567, −2.91356174033816787042561905721, −1.92196364251709850177562557522, 0.42434342038270134009584734092, 1.68749289700133585891229038586, 2.72953202900194826178921398198, 3.63801686204656616510855703063, 5.18462950986039266710484665062, 5.97714703825491855212451215292, 6.21827100939087052478312296004, 7.46842258473211452558348238326, 8.359461415173942912768801803279, 8.865057556756161507676154057160

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