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Properties

Label 2-1680-105.104-c1-0-19
Degree $2$
Conductor $1680$
Sign $0.0759 - 0.997i$
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  + (1.70 + 0.301i)3-s + (−2.04 − 0.902i)5-s + (−1.30 − 2.30i)7-s + (2.81 + 1.02i)9-s + 5.71i·11-s − 3.76·13-s + (−3.21 − 2.15i)15-s + 3.22i·17-s + 0.786i·19-s + (−1.52 − 4.32i)21-s + 1.52·23-s + (3.37 + 3.69i)25-s + (4.49 + 2.60i)27-s + 6.77i·29-s − 1.56i·31-s + ⋯
L(s)  = 1  + (0.984 + 0.174i)3-s + (−0.914 − 0.403i)5-s + (−0.492 − 0.870i)7-s + (0.939 + 0.343i)9-s + 1.72i·11-s − 1.04·13-s + (−0.830 − 0.556i)15-s + 0.783i·17-s + 0.180i·19-s + (−0.333 − 0.942i)21-s + 0.318·23-s + (0.674 + 0.738i)25-s + (0.865 + 0.501i)27-s + 1.25i·29-s − 0.281i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0759 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0759 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.0759 - 0.997i$
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: \((2,\ 1680,\ (\ :1/2),\ 0.0759 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.439226231\)
\(L(\frac12)\) \(\approx\) \(1.439226231\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 0.301i)T \)
5 \( 1 + (2.04 + 0.902i)T \)
7 \( 1 + (1.30 + 2.30i)T \)
good11 \( 1 - 5.71iT - 11T^{2} \)
13 \( 1 + 3.76T + 13T^{2} \)
17 \( 1 - 3.22iT - 17T^{2} \)
19 \( 1 - 0.786iT - 19T^{2} \)
23 \( 1 - 1.52T + 23T^{2} \)
29 \( 1 - 6.77iT - 29T^{2} \)
31 \( 1 + 1.56iT - 31T^{2} \)
37 \( 1 - 8.83iT - 37T^{2} \)
41 \( 1 - 6.65T + 41T^{2} \)
43 \( 1 + 8.85iT - 43T^{2} \)
47 \( 1 + 1.75iT - 47T^{2} \)
53 \( 1 + 0.616T + 53T^{2} \)
59 \( 1 + 6.38T + 59T^{2} \)
61 \( 1 - 14.4iT - 61T^{2} \)
67 \( 1 - 6.26iT - 67T^{2} \)
71 \( 1 - 5.09iT - 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 + 10.5iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + 7.69T + 97T^{2} \)
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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

−9.557733634922196659044741153716, −8.752148640601943206842486502804, −7.84886392830878623244017432006, −7.27560663039021516576434139694, −6.82224154048173352143935826624, −5.06541261111441880074587850625, −4.36575506301214673049953621011, −3.76703037964713759952246083791, −2.68119123082170350448790753594, −1.42737872028764889319433171113, 0.49122222108593035723535042600, 2.47521247279222709859386572501, 3.00087459030611612487718365117, 3.82755007624506560587029842649, 4.95068870791955093158932629481, 6.10052544438147843292946111279, 6.86865918650789996838642118885, 7.84564072365664034362327656275, 8.193983756221428601644512941370, 9.281763275292204025914096888631

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