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Properties

Label 2-264-88.21-c2-0-46
Degree $2$
Conductor $264$
Sign $-0.937 + 0.347i$
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  + (1.48 − 1.34i)2-s − 1.73i·3-s + (0.400 − 3.97i)4-s − 5.07i·5-s + (−2.32 − 2.56i)6-s + 3.53i·7-s + (−4.74 − 6.44i)8-s − 2.99·9-s + (−6.80 − 7.52i)10-s + (−10.5 − 3.03i)11-s + (−6.89 − 0.693i)12-s + 9.79·13-s + (4.73 + 5.23i)14-s − 8.79·15-s + (−15.6 − 3.18i)16-s + 1.66i·17-s + ⋯
L(s)  = 1  + (0.741 − 0.670i)2-s − 0.577i·3-s + (0.100 − 0.994i)4-s − 1.01i·5-s + (−0.387 − 0.428i)6-s + 0.504i·7-s + (−0.593 − 0.805i)8-s − 0.333·9-s + (−0.680 − 0.752i)10-s + (−0.961 − 0.276i)11-s + (−0.574 − 0.0578i)12-s + 0.753·13-s + (0.338 + 0.374i)14-s − 0.586·15-s + (−0.979 − 0.199i)16-s + 0.0977i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $-0.937 + 0.347i$
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: \((2,\ 264,\ (\ :1),\ -0.937 + 0.347i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.367621 - 2.04880i\)
\(L(\frac12)\) \(\approx\) \(0.367621 - 2.04880i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.48 + 1.34i)T \)
3 \( 1 + 1.73iT \)
11 \( 1 + (10.5 + 3.03i)T \)
good5 \( 1 + 5.07iT - 25T^{2} \)
7 \( 1 - 3.53iT - 49T^{2} \)
13 \( 1 - 9.79T + 169T^{2} \)
17 \( 1 - 1.66iT - 289T^{2} \)
19 \( 1 - 26.1T + 361T^{2} \)
23 \( 1 + 7.53T + 529T^{2} \)
29 \( 1 + 3.41T + 841T^{2} \)
31 \( 1 + 35.1T + 961T^{2} \)
37 \( 1 + 58.3iT - 1.36e3T^{2} \)
41 \( 1 + 15.1iT - 1.68e3T^{2} \)
43 \( 1 - 51.1T + 1.84e3T^{2} \)
47 \( 1 - 34.3T + 2.20e3T^{2} \)
53 \( 1 + 91.1iT - 2.80e3T^{2} \)
59 \( 1 - 30.2iT - 3.48e3T^{2} \)
61 \( 1 - 56.5T + 3.72e3T^{2} \)
67 \( 1 - 109. iT - 4.48e3T^{2} \)
71 \( 1 - 23.8T + 5.04e3T^{2} \)
73 \( 1 - 49.2iT - 5.32e3T^{2} \)
79 \( 1 - 15.0iT - 6.24e3T^{2} \)
83 \( 1 - 72.5T + 6.88e3T^{2} \)
89 \( 1 - 13.7T + 7.92e3T^{2} \)
97 \( 1 - 149.T + 9.40e3T^{2} \)
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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

−11.54917620790044713771414678847, −10.66309063156813341994076781646, −9.385373771885862122527682787658, −8.573892928272903524746961546537, −7.30449074136273185457222628550, −5.72575133822556878388241773684, −5.31674456040038791522813614073, −3.77384483144459572062757076689, −2.32821920830174914982755315728, −0.865325937684262661125238027947, 2.81906919089684159346250883647, 3.75871232311806903875740956006, 5.02766609886271120508870334743, 6.06783628216939168637712385124, 7.19828169717875504887554418889, 7.912155047403271912314678559825, 9.270458541005091420636281610241, 10.49207028790373614270745999780, 11.12636480439775829468936441569, 12.20953441573547394059898486444

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