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Properties

Label 2-1320-1320.557-c0-0-3
Degree $2$
Conductor $1320$
Sign $-0.442 + 0.896i$
Analytic cond. $0.658765$
Root an. cond. $0.811643$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.987 − 0.156i)2-s + (−0.453 − 0.891i)3-s + (0.951 − 0.309i)4-s + (0.453 − 0.891i)5-s + (−0.587 − 0.809i)6-s + (−1.76 − 0.896i)7-s + (0.891 − 0.453i)8-s + (−0.587 + 0.809i)9-s + (0.309 − 0.951i)10-s + (0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + (−1.87 − 0.610i)14-s − 1.00·15-s + (0.809 − 0.587i)16-s + (−0.453 + 0.891i)18-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s + (−0.453 − 0.891i)3-s + (0.951 − 0.309i)4-s + (0.453 − 0.891i)5-s + (−0.587 − 0.809i)6-s + (−1.76 − 0.896i)7-s + (0.891 − 0.453i)8-s + (−0.587 + 0.809i)9-s + (0.309 − 0.951i)10-s + (0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + (−1.87 − 0.610i)14-s − 1.00·15-s + (0.809 − 0.587i)16-s + (−0.453 + 0.891i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.442 + 0.896i$
Analytic conductor: \(0.658765\)
Root analytic conductor: \(0.811643\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :0),\ -0.442 + 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.560165464\)
\(L(\frac12)\) \(\approx\) \(1.560165464\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.987 + 0.156i)T \)
3 \( 1 + (0.453 + 0.891i)T \)
5 \( 1 + (-0.453 + 0.891i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (1.76 + 0.896i)T + (0.587 + 0.809i)T^{2} \)
13 \( 1 + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-0.280 - 0.863i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (-1.59 + 0.253i)T + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (-0.297 + 0.0966i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 - 1.58i)T + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.610 - 0.0966i)T + (0.951 + 0.309i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.278 + 1.76i)T + (-0.951 + 0.309i)T^{2} \)
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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

−9.805313314455003515468603795635, −8.840340843444649151446490123065, −7.44073224271043118988215446029, −6.91745825198739960847028094300, −6.22298217024864011537210189550, −5.51505068281514958169530478183, −4.44268015443659088186460750946, −3.54492206022855051899593080813, −2.27446277198320493719835945664, −1.05410395787597964012478253207, 2.49182309999008282934709298529, 3.32015434295935359615211180522, 3.83720917819246181384200349883, 5.26454574345341886525984975955, 6.07067171417993449585737642945, 6.30600975392338890198525903251, 7.16437776054527644806100217752, 8.725948124830310537024524316551, 9.437202644754023649675549918229, 10.21318972340847034020890940279

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