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Properties

Label 2-264-88.21-c2-0-31
Degree $2$
Conductor $264$
Sign $0.902 + 0.431i$
Analytic cond. $7.19347$
Root an. cond. $2.68206$
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  + (1.98 + 0.209i)2-s − 1.73i·3-s + (3.91 + 0.833i)4-s + 4.86i·5-s + (0.362 − 3.44i)6-s − 11.9i·7-s + (7.60 + 2.47i)8-s − 2.99·9-s + (−1.01 + 9.66i)10-s + (1.44 − 10.9i)11-s + (1.44 − 6.77i)12-s + 20.4·13-s + (2.50 − 23.8i)14-s + 8.41·15-s + (14.6 + 6.52i)16-s + 26.3i·17-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s − 0.577i·3-s + (0.978 + 0.208i)4-s + 0.972i·5-s + (0.0604 − 0.574i)6-s − 1.71i·7-s + (0.950 + 0.309i)8-s − 0.333·9-s + (−0.101 + 0.966i)10-s + (0.130 − 0.991i)11-s + (0.120 − 0.564i)12-s + 1.57·13-s + (0.179 − 1.70i)14-s + 0.561·15-s + (0.913 + 0.407i)16-s + 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.902 + 0.431i$
Analytic conductor: \(7.19347\)
Root analytic conductor: \(2.68206\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :1),\ 0.902 + 0.431i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.99857 - 0.680205i\)
\(L(\frac12)\) \(\approx\) \(2.99857 - 0.680205i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.98 - 0.209i)T \)
3 \( 1 + 1.73iT \)
11 \( 1 + (-1.44 + 10.9i)T \)
good5 \( 1 - 4.86iT - 25T^{2} \)
7 \( 1 + 11.9iT - 49T^{2} \)
13 \( 1 - 20.4T + 169T^{2} \)
17 \( 1 - 26.3iT - 289T^{2} \)
19 \( 1 - 12.5T + 361T^{2} \)
23 \( 1 + 35.3T + 529T^{2} \)
29 \( 1 + 13.2T + 841T^{2} \)
31 \( 1 + 19.6T + 961T^{2} \)
37 \( 1 + 9.27iT - 1.36e3T^{2} \)
41 \( 1 + 13.7iT - 1.68e3T^{2} \)
43 \( 1 + 23.0T + 1.84e3T^{2} \)
47 \( 1 + 63.7T + 2.20e3T^{2} \)
53 \( 1 - 21.7iT - 2.80e3T^{2} \)
59 \( 1 - 66.8iT - 3.48e3T^{2} \)
61 \( 1 + 50.9T + 3.72e3T^{2} \)
67 \( 1 - 87.6iT - 4.48e3T^{2} \)
71 \( 1 - 13.1T + 5.04e3T^{2} \)
73 \( 1 + 47.7iT - 5.32e3T^{2} \)
79 \( 1 - 39.2iT - 6.24e3T^{2} \)
83 \( 1 - 60.8T + 6.88e3T^{2} \)
89 \( 1 - 25.2T + 7.92e3T^{2} \)
97 \( 1 - 107.T + 9.40e3T^{2} \)
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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

−11.58281341257186310622701524983, −10.84330068682814946517711559142, −10.39398029488445109337822291742, −8.301996141437233812941547075961, −7.47972628505848687467822195415, −6.49427193496033477488552381252, −5.91591267823991532025541611652, −3.92294279163346187479340503505, −3.41655430206515744042203182545, −1.45633814889725968994228600975, 1.88788375720237814857654522709, 3.35236740932827319759347145487, 4.73816947986262970774898893620, 5.39271955833466699118386179151, 6.36364475123699846877665186356, 7.993923609737516896970276689449, 9.077677244397750077877739449503, 9.795744808659072658714714674721, 11.30941099406985816547219492645, 11.89751428264807424555237427539

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