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Properties

Label 2-1320-88.43-c1-0-33
Degree $2$
Conductor $1320$
Sign $0.183 - 0.983i$
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  + (−1.39 + 0.239i)2-s + 3-s + (1.88 − 0.667i)4-s + i·5-s + (−1.39 + 0.239i)6-s + 3.76·7-s + (−2.46 + 1.38i)8-s + 9-s + (−0.239 − 1.39i)10-s + (2.54 + 2.12i)11-s + (1.88 − 0.667i)12-s − 6.13·13-s + (−5.25 + 0.902i)14-s + i·15-s + (3.10 − 2.51i)16-s + 3.13i·17-s + ⋯
L(s)  = 1  + (−0.985 + 0.169i)2-s + 0.577·3-s + (0.942 − 0.333i)4-s + 0.447i·5-s + (−0.569 + 0.0977i)6-s + 1.42·7-s + (−0.872 + 0.488i)8-s + 0.333·9-s + (−0.0757 − 0.440i)10-s + (0.768 + 0.640i)11-s + (0.544 − 0.192i)12-s − 1.70·13-s + (−1.40 + 0.241i)14-s + 0.258i·15-s + (0.777 − 0.629i)16-s + 0.759i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.183 - 0.983i$
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: \((2,\ 1320,\ (\ :1/2),\ 0.183 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.420371560\)
\(L(\frac12)\) \(\approx\) \(1.420371560\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.239i)T \)
3 \( 1 - T \)
5 \( 1 - iT \)
11 \( 1 + (-2.54 - 2.12i)T \)
good7 \( 1 - 3.76T + 7T^{2} \)
13 \( 1 + 6.13T + 13T^{2} \)
17 \( 1 - 3.13iT - 17T^{2} \)
19 \( 1 - 2.97iT - 19T^{2} \)
23 \( 1 - 7.10iT - 23T^{2} \)
29 \( 1 + 0.755T + 29T^{2} \)
31 \( 1 - 7.99iT - 31T^{2} \)
37 \( 1 + 11.4iT - 37T^{2} \)
41 \( 1 - 1.50iT - 41T^{2} \)
43 \( 1 + 9.18iT - 43T^{2} \)
47 \( 1 - 5.03iT - 47T^{2} \)
53 \( 1 - 4.19iT - 53T^{2} \)
59 \( 1 + 3.18T + 59T^{2} \)
61 \( 1 + 1.43T + 61T^{2} \)
67 \( 1 - 4.13T + 67T^{2} \)
71 \( 1 + 15.7iT - 71T^{2} \)
73 \( 1 - 5.45iT - 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 + 0.0214iT - 83T^{2} \)
89 \( 1 - 0.0462T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−9.642373920189843250857597398010, −9.060583772397719510373890246858, −8.101071229989125794232416635950, −7.47717918869771391051539076571, −7.04330258173776450351481956585, −5.75049607323080686080173972088, −4.79291075353590432403697357469, −3.57008541130688818886191255495, −2.17659215352338232116926878300, −1.59624353165687967230452311001, 0.76451362001611788950178334439, 2.02845190420271618292636470239, 2.85023641722864356131108107777, 4.34603048007559882399910319315, 5.07621844072409701104377868186, 6.46905523560525239735755275490, 7.32937384003932096665753847948, 8.083000649507754349545223155527, 8.567362748356423820546859901053, 9.417681460171565988153228354115

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