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Properties

Label 2-1680-105.104-c1-0-56
Degree $2$
Conductor $1680$
Sign $0.383 - 0.923i$
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  + (0.726 + 1.57i)3-s + (2.20 + 0.341i)5-s + (2.06 + 1.64i)7-s + (−1.94 + 2.28i)9-s − 1.06i·11-s + 4.82·13-s + (1.06 + 3.72i)15-s − 7.89i·17-s + 4.02i·19-s + (−1.09 + 4.45i)21-s + 5.69·23-s + (4.76 + 1.50i)25-s + (−5.00 − 1.40i)27-s − 2.00i·29-s + 4.89i·31-s + ⋯
L(s)  = 1  + (0.419 + 0.907i)3-s + (0.988 + 0.152i)5-s + (0.781 + 0.623i)7-s + (−0.648 + 0.761i)9-s − 0.320i·11-s + 1.33·13-s + (0.275 + 0.961i)15-s − 1.91i·17-s + 0.922i·19-s + (−0.238 + 0.971i)21-s + 1.18·23-s + (0.953 + 0.301i)25-s + (−0.962 − 0.269i)27-s − 0.372i·29-s + 0.879i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.383 - 0.923i$
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: \((2,\ 1680,\ (\ :1/2),\ 0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.765276327\)
\(L(\frac12)\) \(\approx\) \(2.765276327\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.726 - 1.57i)T \)
5 \( 1 + (-2.20 - 0.341i)T \)
7 \( 1 + (-2.06 - 1.64i)T \)
good11 \( 1 + 1.06iT - 11T^{2} \)
13 \( 1 - 4.82T + 13T^{2} \)
17 \( 1 + 7.89iT - 17T^{2} \)
19 \( 1 - 4.02iT - 19T^{2} \)
23 \( 1 - 5.69T + 23T^{2} \)
29 \( 1 + 2.00iT - 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 + 2.56iT - 37T^{2} \)
41 \( 1 + 5.08T + 41T^{2} \)
43 \( 1 + 6.15iT - 43T^{2} \)
47 \( 1 - 2.27iT - 47T^{2} \)
53 \( 1 + 9.84T + 53T^{2} \)
59 \( 1 + 5.87T + 59T^{2} \)
61 \( 1 + 7.02iT - 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 + 0.0512iT - 71T^{2} \)
73 \( 1 - 2.86T + 73T^{2} \)
79 \( 1 + 7.00T + 79T^{2} \)
83 \( 1 + 7.59iT - 83T^{2} \)
89 \( 1 + 9.72T + 89T^{2} \)
97 \( 1 + 1.77T + 97T^{2} \)
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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

−9.379479896272119461730276114581, −8.853317900985551569314862944419, −8.241402902878830685853127866369, −7.13380088138917470129285002222, −6.01425976909947988663418466780, −5.33919326105383795033629793356, −4.71330372498185591817119852252, −3.40730093297548724043020726274, −2.66029630981537816956554138404, −1.47907873380457678932410445595, 1.22002270664705749035067955429, 1.75087385023808356883501379530, 3.02154801356955635159777684469, 4.12811647117225951300689911932, 5.21541007739403230304830624334, 6.21798916717797040870999186072, 6.67487122620788161260959302487, 7.70078777557304392486922664229, 8.460171524886276101198535925857, 8.943726807962767847725091889172

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