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Properties

Label 2-2160-15.14-c2-0-52
Degree $2$
Conductor $2160$
Sign $0.948 - 0.316i$
Analytic cond. $58.8557$
Root an. cond. $7.67174$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.58 + 4.74i)5-s − 3i·7-s + 9.48i·11-s − 21i·13-s + 12.6·17-s + 31·19-s − 22.1·23-s + (−20 + 15.0i)25-s + 47.4i·29-s + 16·31-s + (14.2 − 4.74i)35-s − 27i·37-s − 47.4i·41-s − 48i·43-s − 12.6·47-s + ⋯
L(s)  = 1  + (0.316 + 0.948i)5-s − 0.428i·7-s + 0.862i·11-s − 1.61i·13-s + 0.744·17-s + 1.63·19-s − 0.962·23-s + (−0.800 + 0.600i)25-s + 1.63i·29-s + 0.516·31-s + (0.406 − 0.135i)35-s − 0.729i·37-s − 1.15i·41-s − 1.11i·43-s − 0.269·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.948 - 0.316i$
Analytic conductor: \(58.8557\)
Root analytic conductor: \(7.67174\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1),\ 0.948 - 0.316i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.303868719\)
\(L(\frac12)\) \(\approx\) \(2.303868719\)
\(L(2)\) 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 \( 1 \)
5 \( 1 + (-1.58 - 4.74i)T \)
good7 \( 1 + 3iT - 49T^{2} \)
11 \( 1 - 9.48iT - 121T^{2} \)
13 \( 1 + 21iT - 169T^{2} \)
17 \( 1 - 12.6T + 289T^{2} \)
19 \( 1 - 31T + 361T^{2} \)
23 \( 1 + 22.1T + 529T^{2} \)
29 \( 1 - 47.4iT - 841T^{2} \)
31 \( 1 - 16T + 961T^{2} \)
37 \( 1 + 27iT - 1.36e3T^{2} \)
41 \( 1 + 47.4iT - 1.68e3T^{2} \)
43 \( 1 + 48iT - 1.84e3T^{2} \)
47 \( 1 + 12.6T + 2.20e3T^{2} \)
53 \( 1 - 41.1T + 2.80e3T^{2} \)
59 \( 1 + 37.9iT - 3.48e3T^{2} \)
61 \( 1 + T + 3.72e3T^{2} \)
67 \( 1 - 21iT - 4.48e3T^{2} \)
71 \( 1 + 28.4iT - 5.04e3T^{2} \)
73 \( 1 + 27iT - 5.32e3T^{2} \)
79 \( 1 - T + 6.24e3T^{2} \)
83 \( 1 - 110.T + 6.88e3T^{2} \)
89 \( 1 - 113. iT - 7.92e3T^{2} \)
97 \( 1 - 93iT - 9.40e3T^{2} \)
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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

−9.049813919030524521190816555744, −7.82071360774168413335366973157, −7.47455106581811435814665213821, −6.74216277398793443786523585711, −5.61462112950724982786437976544, −5.21606256385350892610978870699, −3.75819838196827559920334672383, −3.18242905278834579060466774321, −2.12437926927513239643605328073, −0.822736455912634815373562912243, 0.817008663883462826980026789393, 1.78980819500163831422210000937, 2.94895449791701500892313100055, 4.08253635947917157952260224222, 4.83036610195798083841593859173, 5.81473184084430859636675450310, 6.21438062759510482786683276719, 7.44431693313266553502896762038, 8.215187764869712416736522598729, 8.829304674493896433473650371771

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