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Properties

Label 2-1680-12.11-c1-0-31
Degree $2$
Conductor $1680$
Sign $0.992 + 0.120i$
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  + (1.71 + 0.209i)3-s − i·5-s + i·7-s + (2.91 + 0.719i)9-s − 1.73·11-s + 3.59·13-s + (0.209 − 1.71i)15-s + 1.58i·17-s − 7.56i·19-s + (−0.209 + 1.71i)21-s + 5.40·23-s − 25-s + (4.85 + 1.84i)27-s − 2.80i·29-s + 0.682i·31-s + ⋯
L(s)  = 1  + (0.992 + 0.120i)3-s − 0.447i·5-s + 0.377i·7-s + (0.970 + 0.239i)9-s − 0.523·11-s + 0.996·13-s + (0.0540 − 0.443i)15-s + 0.383i·17-s − 1.73i·19-s + (−0.0456 + 0.375i)21-s + 1.12·23-s − 0.200·25-s + (0.934 + 0.355i)27-s − 0.520i·29-s + 0.122i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.992 + 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.635904706\)
\(L(\frac12)\) \(\approx\) \(2.635904706\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.209i)T \)
5 \( 1 + iT \)
7 \( 1 - iT \)
good11 \( 1 + 1.73T + 11T^{2} \)
13 \( 1 - 3.59T + 13T^{2} \)
17 \( 1 - 1.58iT - 17T^{2} \)
19 \( 1 + 7.56iT - 19T^{2} \)
23 \( 1 - 5.40T + 23T^{2} \)
29 \( 1 + 2.80iT - 29T^{2} \)
31 \( 1 - 0.682iT - 31T^{2} \)
37 \( 1 - 8.04T + 37T^{2} \)
41 \( 1 - 7.40iT - 41T^{2} \)
43 \( 1 - 5.40iT - 43T^{2} \)
47 \( 1 + 1.72T + 47T^{2} \)
53 \( 1 + 0.990iT - 53T^{2} \)
59 \( 1 + 7.71T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 0.285iT - 67T^{2} \)
71 \( 1 - 1.86T + 71T^{2} \)
73 \( 1 - 9.56T + 73T^{2} \)
79 \( 1 + 9.68iT - 79T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 - 12.5iT - 89T^{2} \)
97 \( 1 + 4.71T + 97T^{2} \)
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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

−9.256991780653962091918437981659, −8.534863858921500482801472072145, −8.025052852090088359702259269165, −7.05557251973336728297760626038, −6.17225489283347890786007761276, −5.02645544006750995117754018099, −4.35365925563537221984679444579, −3.20563094530210770697302034248, −2.44850704110050492001286391117, −1.12258009097653561621670714899, 1.22789196489058653716696450710, 2.42210058972937053970923226308, 3.42689404194258765016102033226, 4.02087206053895333968619310047, 5.27543792695685625337854767672, 6.30444440710422361544801650513, 7.14912315044027159341425921475, 7.82631087303824981805282688875, 8.500658400200254448153136636249, 9.302893481014672675186313961955

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