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Properties

Label 2-1320-88.43-c1-0-62
Degree $2$
Conductor $1320$
Sign $0.0661 - 0.997i$
Analytic cond. $10.5402$
Root an. cond. $3.24657$
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  + (0.492 + 1.32i)2-s + 3-s + (−1.51 + 1.30i)4-s + i·5-s + (0.492 + 1.32i)6-s + 4.38·7-s + (−2.47 − 1.36i)8-s + 9-s + (−1.32 + 0.492i)10-s + (3.00 − 1.40i)11-s + (−1.51 + 1.30i)12-s + 3.63·13-s + (2.16 + 5.81i)14-s + i·15-s + (0.589 − 3.95i)16-s + 3.74i·17-s + ⋯
L(s)  = 1  + (0.348 + 0.937i)2-s + 0.577·3-s + (−0.757 + 0.652i)4-s + 0.447i·5-s + (0.201 + 0.541i)6-s + 1.65·7-s + (−0.875 − 0.482i)8-s + 0.333·9-s + (−0.419 + 0.155i)10-s + (0.905 − 0.423i)11-s + (−0.437 + 0.376i)12-s + 1.00·13-s + (0.577 + 1.55i)14-s + 0.258i·15-s + (0.147 − 0.989i)16-s + 0.908i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0661 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0661 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.0661 - 0.997i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ 0.0661 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.886223823\)
\(L(\frac12)\) \(\approx\) \(2.886223823\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.492 - 1.32i)T \)
3 \( 1 - T \)
5 \( 1 - iT \)
11 \( 1 + (-3.00 + 1.40i)T \)
good7 \( 1 - 4.38T + 7T^{2} \)
13 \( 1 - 3.63T + 13T^{2} \)
17 \( 1 - 3.74iT - 17T^{2} \)
19 \( 1 + 2.02iT - 19T^{2} \)
23 \( 1 + 8.46iT - 23T^{2} \)
29 \( 1 + 3.70T + 29T^{2} \)
31 \( 1 + 5.04iT - 31T^{2} \)
37 \( 1 + 4.94iT - 37T^{2} \)
41 \( 1 - 8.85iT - 41T^{2} \)
43 \( 1 - 6.69iT - 43T^{2} \)
47 \( 1 - 8.37iT - 47T^{2} \)
53 \( 1 - 2.58iT - 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 8.00T + 61T^{2} \)
67 \( 1 + 9.62T + 67T^{2} \)
71 \( 1 + 8.09iT - 71T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 + 3.48T + 79T^{2} \)
83 \( 1 - 1.85iT - 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 - 16.3T + 97T^{2} \)
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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

−9.442941451348117881534509479917, −8.698692909786150461065481867605, −8.153213384220421694900786714239, −7.54769940201266552822707826605, −6.42990974235860148759955630854, −5.89834154864253438961160963010, −4.52422926619793603269912787263, −4.12972473390670487922770733601, −2.90405010215164497708955553991, −1.43193502227071984508856263159, 1.38243071694864897195084688247, 1.80866084177175970901951459510, 3.37457757543432882869263954603, 4.14025210493837136157960729284, 4.99546053311972492931751933110, 5.71497222296136527192327257508, 7.16274154900434598537579453036, 8.068497886911741797664698968337, 8.880614356114584410402291819981, 9.285314889326224918054606789309

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