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Properties

Label 2-720-144.11-c1-0-13
Degree $2$
Conductor $720$
Sign $-0.969 + 0.243i$
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  + (1.26 + 0.630i)2-s + (−1.02 + 1.39i)3-s + (1.20 + 1.59i)4-s + (−0.965 − 0.258i)5-s + (−2.17 + 1.12i)6-s + (−0.936 + 1.62i)7-s + (0.516 + 2.78i)8-s + (−0.906 − 2.85i)9-s + (−1.05 − 0.937i)10-s + (−1.54 + 0.412i)11-s + (−3.46 + 0.0486i)12-s + (−5.44 − 1.45i)13-s + (−2.20 + 1.46i)14-s + (1.34 − 1.08i)15-s + (−1.10 + 3.84i)16-s + 1.94i·17-s + ⋯
L(s)  = 1  + (0.894 + 0.446i)2-s + (−0.590 + 0.806i)3-s + (0.601 + 0.798i)4-s + (−0.431 − 0.115i)5-s + (−0.888 + 0.458i)6-s + (−0.354 + 0.613i)7-s + (0.182 + 0.983i)8-s + (−0.302 − 0.953i)9-s + (−0.334 − 0.296i)10-s + (−0.464 + 0.124i)11-s + (−0.999 + 0.0140i)12-s + (−1.50 − 0.404i)13-s + (−0.590 + 0.390i)14-s + (0.348 − 0.280i)15-s + (−0.275 + 0.961i)16-s + 0.470i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.969 + 0.243i$
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: \((2,\ 720,\ (\ :1/2),\ -0.969 + 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.135723 - 1.09670i\)
\(L(\frac12)\) \(\approx\) \(0.135723 - 1.09670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 0.630i)T \)
3 \( 1 + (1.02 - 1.39i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
good7 \( 1 + (0.936 - 1.62i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.54 - 0.412i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (5.44 + 1.45i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 1.94iT - 17T^{2} \)
19 \( 1 + (-1.56 - 1.56i)T + 19iT^{2} \)
23 \( 1 + (-1.20 + 0.695i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.12 - 1.37i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-1.40 + 0.812i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.45 - 3.45i)T + 37iT^{2} \)
41 \( 1 + (3.66 + 6.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.930 - 3.47i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.48 - 2.58i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.95 - 4.95i)T - 53iT^{2} \)
59 \( 1 + (-1.86 + 6.96i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.81 + 6.77i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.94 - 11.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.65iT - 71T^{2} \)
73 \( 1 - 15.9iT - 73T^{2} \)
79 \( 1 + (-13.5 - 7.83i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.82 + 6.81i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + (-2.20 + 3.82i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.03200232689662436225340903541, −10.10413065874786178187824676015, −9.223116899635135400803035743854, −8.109793932648310512223762529264, −7.24848850982499226582060367119, −6.18218796700517784444917375705, −5.34071476603464892749313233549, −4.71635464210161719509280636923, −3.62054333846980794849623382907, −2.61477388281658666439002297886, 0.42780095446834619250038166928, 2.11522933152478067993479330883, 3.20719873773672413677521656709, 4.57863474050371606220928786027, 5.26321955951245681278867469107, 6.39969467239674161699739187241, 7.22192273955838565389871257216, 7.68026273151006413488837148023, 9.370707777623545832393593556989, 10.27415044509909994315037099900

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