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Properties

Label 2-1320-55.43-c1-0-29
Degree $2$
Conductor $1320$
Sign $-0.298 + 0.954i$
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  + (−0.707 − 0.707i)3-s + (1.17 − 1.90i)5-s + (0.360 + 0.360i)7-s + 1.00i·9-s + (2.30 − 2.38i)11-s + (3.45 − 3.45i)13-s + (−2.17 + 0.514i)15-s + (−2.46 − 2.46i)17-s − 8.10·19-s − 0.510i·21-s + (5.54 + 5.54i)23-s + (−2.23 − 4.47i)25-s + (0.707 − 0.707i)27-s + 2.53·29-s + 4.86·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.525 − 0.850i)5-s + (0.136 + 0.136i)7-s + 0.333i·9-s + (0.694 − 0.719i)11-s + (0.959 − 0.959i)13-s + (−0.561 + 0.132i)15-s + (−0.598 − 0.598i)17-s − 1.86·19-s − 0.111i·21-s + (1.15 + 1.15i)23-s + (−0.447 − 0.894i)25-s + (0.136 − 0.136i)27-s + 0.470·29-s + 0.873·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.298 + 0.954i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ -0.298 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.526026658\)
\(L(\frac12)\) \(\approx\) \(1.526026658\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.17 + 1.90i)T \)
11 \( 1 + (-2.30 + 2.38i)T \)
good7 \( 1 + (-0.360 - 0.360i)T + 7iT^{2} \)
13 \( 1 + (-3.45 + 3.45i)T - 13iT^{2} \)
17 \( 1 + (2.46 + 2.46i)T + 17iT^{2} \)
19 \( 1 + 8.10T + 19T^{2} \)
23 \( 1 + (-5.54 - 5.54i)T + 23iT^{2} \)
29 \( 1 - 2.53T + 29T^{2} \)
31 \( 1 - 4.86T + 31T^{2} \)
37 \( 1 + (-2.42 + 2.42i)T - 37iT^{2} \)
41 \( 1 - 6.16iT - 41T^{2} \)
43 \( 1 + (-0.334 + 0.334i)T - 43iT^{2} \)
47 \( 1 + (5.54 - 5.54i)T - 47iT^{2} \)
53 \( 1 + (3.90 + 3.90i)T + 53iT^{2} \)
59 \( 1 + 13.9iT - 59T^{2} \)
61 \( 1 + 9.57iT - 61T^{2} \)
67 \( 1 + (0.106 - 0.106i)T - 67iT^{2} \)
71 \( 1 + 4.43T + 71T^{2} \)
73 \( 1 + (-0.961 + 0.961i)T - 73iT^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + (4.08 - 4.08i)T - 83iT^{2} \)
89 \( 1 + 8.18iT - 89T^{2} \)
97 \( 1 + (-0.726 + 0.726i)T - 97iT^{2} \)
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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

−9.246406650466203779550890670539, −8.543304607520541691494913891358, −8.014444185799298427208430380803, −6.60795880472257647296380623100, −6.16598656043220857291280824745, −5.25121555728761600909315995032, −4.42847462911850430425463768091, −3.14603968334218092430152589143, −1.76463275873027075418668153726, −0.69921122796526655616014021058, 1.57388213837256169480853808579, 2.69491404559562459474734775948, 4.12717033840580512894819156556, 4.50108265246207342581714819742, 6.01287579847614748694541533111, 6.52059061744149754935733142936, 7.06134973224619719669175058280, 8.571916328197980211009331788493, 8.995266804928613483048258276249, 10.08421362440549601933814215170

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