Properties

Label 2-1680-5.4-c1-0-15
Degree $2$
Conductor $1680$
Sign $0.139 - 0.990i$
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  + i·3-s + (−0.311 + 2.21i)5-s i·7-s − 9-s + 5.05·11-s + 3.37i·13-s + (−2.21 − 0.311i)15-s − 7.18i·17-s + 8.23·19-s + 21-s + 6.23i·23-s + (−4.80 − 1.37i)25-s i·27-s + 2·29-s + 4.62·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.139 + 0.990i)5-s − 0.377i·7-s − 0.333·9-s + 1.52·11-s + 0.936i·13-s + (−0.571 − 0.0803i)15-s − 1.74i·17-s + 1.88·19-s + 0.218·21-s + 1.30i·23-s + (−0.961 − 0.275i)25-s − 0.192i·27-s + 0.371·29-s + 0.830·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.139 - 0.990i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.845173066\)
\(L(\frac12)\) \(\approx\) \(1.845173066\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (0.311 - 2.21i)T \)
7 \( 1 + iT \)
good11 \( 1 - 5.05T + 11T^{2} \)
13 \( 1 - 3.37iT - 13T^{2} \)
17 \( 1 + 7.18iT - 17T^{2} \)
19 \( 1 - 8.23T + 19T^{2} \)
23 \( 1 - 6.23iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 4.62T + 31T^{2} \)
37 \( 1 - 4.85iT - 37T^{2} \)
41 \( 1 + 3.37T + 41T^{2} \)
43 \( 1 + 1.24iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 4.62iT - 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 - 0.488T + 61T^{2} \)
67 \( 1 - 3.61iT - 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 16.2iT - 73T^{2} \)
79 \( 1 - 1.24T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + 6.99T + 89T^{2} \)
97 \( 1 - 8.23iT - 97T^{2} \)
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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

−9.575199599485730602064472112723, −9.077758646062534145202306931835, −7.76933420356356800007574736206, −7.05684539044720951536632419713, −6.52705833955452526707345737382, −5.39747373645735920743117779802, −4.43686246709029978677315629674, −3.56202432856041743610982345722, −2.86349811258786289621682911177, −1.27109624379753640853342390219, 0.847910724087556684829637899944, 1.72951325100819977894398342628, 3.18745272688949554926810339142, 4.13077921822599670970705716554, 5.15601315600946275554836027872, 5.98974356769463674405524464335, 6.64712953142828354371986188107, 7.82625289499980702418707607753, 8.320086037480194555609848024004, 9.053968974486423362052174591297

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