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Properties

Label 2-720-144.11-c1-0-21
Degree $2$
Conductor $720$
Sign $-0.943 + 0.330i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
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  + (0.796 + 1.16i)2-s + (0.727 + 1.57i)3-s + (−0.730 + 1.86i)4-s + (−0.965 − 0.258i)5-s + (−1.25 + 2.10i)6-s + (−0.518 + 0.897i)7-s + (−2.75 + 0.629i)8-s + (−1.94 + 2.28i)9-s + (−0.467 − 1.33i)10-s + (−1.55 + 0.417i)11-s + (−3.45 + 0.206i)12-s + (1.74 + 0.466i)13-s + (−1.46 + 0.109i)14-s + (−0.295 − 1.70i)15-s + (−2.93 − 2.72i)16-s − 1.18i·17-s + ⋯
L(s)  = 1  + (0.563 + 0.826i)2-s + (0.420 + 0.907i)3-s + (−0.365 + 0.930i)4-s + (−0.431 − 0.115i)5-s + (−0.513 + 0.858i)6-s + (−0.195 + 0.339i)7-s + (−0.974 + 0.222i)8-s + (−0.647 + 0.762i)9-s + (−0.147 − 0.422i)10-s + (−0.469 + 0.125i)11-s + (−0.998 + 0.0594i)12-s + (0.483 + 0.129i)13-s + (−0.390 + 0.0292i)14-s + (−0.0764 − 0.440i)15-s + (−0.733 − 0.680i)16-s − 0.286i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.943 + 0.330i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.943 + 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259740 - 1.52603i\)
\(L(\frac12)\) \(\approx\) \(0.259740 - 1.52603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.796 - 1.16i)T \)
3 \( 1 + (-0.727 - 1.57i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
good7 \( 1 + (0.518 - 0.897i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.55 - 0.417i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.74 - 0.466i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 1.18iT - 17T^{2} \)
19 \( 1 + (-1.51 - 1.51i)T + 19iT^{2} \)
23 \( 1 + (3.38 - 1.95i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.72 + 0.461i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-0.697 + 0.402i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.03 + 2.03i)T + 37iT^{2} \)
41 \( 1 + (-2.35 - 4.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.76 - 10.3i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.26 - 2.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.53 + 4.53i)T - 53iT^{2} \)
59 \( 1 + (1.94 - 7.25i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.829 + 3.09i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.0601 + 0.224i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 6.66iT - 71T^{2} \)
73 \( 1 - 0.623iT - 73T^{2} \)
79 \( 1 + (-14.0 - 8.12i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.34 - 5.02i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 3.14T + 89T^{2} \)
97 \( 1 + (5.74 - 9.94i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.92096881644197503304457084749, −9.761540346712551922882994321888, −9.079250077055106303010395176380, −8.151736239440276395883245743681, −7.62536876989104448974147818562, −6.29572158826105875817550834456, −5.41979724807518581657751577365, −4.52339401061617041779226236768, −3.66061672058309235041986282953, −2.68638400966389072434354889165, 0.62541612649450787533471880047, 2.10074929999079360194269301492, 3.19412161305161251443028835788, 4.03320700936759691057045658608, 5.39326371829480953093525035756, 6.35354644746223214443530891502, 7.25077981286101358189637330501, 8.301993186838623624334913851094, 9.034580018298683842173297224901, 10.19649217028098823088531073033

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