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Properties

Label 2-1320-8.5-c1-0-53
Degree $2$
Conductor $1320$
Sign $0.129 + 0.991i$
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  + (−0.653 + 1.25i)2-s − i·3-s + (−1.14 − 1.63i)4-s + i·5-s + (1.25 + 0.653i)6-s − 1.00·7-s + (2.80 − 0.365i)8-s − 9-s + (−1.25 − 0.653i)10-s − i·11-s + (−1.63 + 1.14i)12-s − 0.768i·13-s + (0.659 − 1.26i)14-s + 15-s + (−1.37 + 3.75i)16-s − 1.00·17-s + ⋯
L(s)  = 1  + (−0.462 + 0.886i)2-s − 0.577i·3-s + (−0.572 − 0.819i)4-s + 0.447i·5-s + (0.512 + 0.266i)6-s − 0.381·7-s + (0.991 − 0.129i)8-s − 0.333·9-s + (−0.396 − 0.206i)10-s − 0.301i·11-s + (−0.473 + 0.330i)12-s − 0.213i·13-s + (0.176 − 0.338i)14-s + 0.258·15-s + (−0.343 + 0.939i)16-s − 0.244·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.129 + 0.991i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (661, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ 0.129 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6163310939\)
\(L(\frac12)\) \(\approx\) \(0.6163310939\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.653 - 1.25i)T \)
3 \( 1 + iT \)
5 \( 1 - iT \)
11 \( 1 + iT \)
good7 \( 1 + 1.00T + 7T^{2} \)
13 \( 1 + 0.768iT - 13T^{2} \)
17 \( 1 + 1.00T + 17T^{2} \)
19 \( 1 - 2.21iT - 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 10.3iT - 29T^{2} \)
31 \( 1 + 2.41T + 31T^{2} \)
37 \( 1 + 5.94iT - 37T^{2} \)
41 \( 1 - 1.77T + 41T^{2} \)
43 \( 1 - 6.83iT - 43T^{2} \)
47 \( 1 + 9.30T + 47T^{2} \)
53 \( 1 + 5.05iT - 53T^{2} \)
59 \( 1 + 9.83iT - 59T^{2} \)
61 \( 1 + 6.85iT - 61T^{2} \)
67 \( 1 - 0.795iT - 67T^{2} \)
71 \( 1 + 4.10T + 71T^{2} \)
73 \( 1 - 2.11T + 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 + 5.98T + 89T^{2} \)
97 \( 1 - 5.08T + 97T^{2} \)
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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

−9.431615349569800089226437751159, −8.372445835331788469192946215165, −7.82775954796306719875058681767, −6.98393167635102044906647010981, −6.27082082003462364541155390537, −5.66179366756950182143952280529, −4.48243290598094632478576548860, −3.24650265896487068003232848509, −1.86738269980929789144288532205, −0.31344671582590831790171727126, 1.36112331242017641296543883199, 2.72003826214294677382247615112, 3.61004495188926269764693044769, 4.59749579185351313382635773818, 5.26423764744566987336158185650, 6.68633369929847969062918151213, 7.54884915373451276223326558568, 8.723376611391209082555308133461, 8.994280378843542514159502851262, 9.854760295136122448347544246900

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