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Properties

Label 2-264-88.51-c1-0-4
Degree $2$
Conductor $264$
Sign $0.986 - 0.163i$
Analytic cond. $2.10805$
Root an. cond. $1.45191$
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  + (0.445 − 1.34i)2-s + (−0.309 + 0.951i)3-s + (−1.60 − 1.19i)4-s + (1.48 + 2.04i)5-s + (1.13 + 0.838i)6-s + (1.07 + 3.30i)7-s + (−2.31 + 1.62i)8-s + (−0.809 − 0.587i)9-s + (3.39 − 1.08i)10-s + (2.55 + 2.11i)11-s + (1.63 − 1.15i)12-s + (−1.89 − 1.37i)13-s + (4.90 + 0.0294i)14-s + (−2.39 + 0.779i)15-s + (1.14 + 3.83i)16-s + (−2.12 − 2.92i)17-s + ⋯
L(s)  = 1  + (0.314 − 0.949i)2-s + (−0.178 + 0.549i)3-s + (−0.801 − 0.597i)4-s + (0.663 + 0.912i)5-s + (0.465 + 0.342i)6-s + (0.405 + 1.24i)7-s + (−0.819 + 0.573i)8-s + (−0.269 − 0.195i)9-s + (1.07 − 0.342i)10-s + (0.770 + 0.637i)11-s + (0.471 − 0.333i)12-s + (−0.525 − 0.381i)13-s + (1.31 + 0.00787i)14-s + (−0.619 + 0.201i)15-s + (0.286 + 0.958i)16-s + (−0.516 − 0.710i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.986 - 0.163i$
Analytic conductor: \(2.10805\)
Root analytic conductor: \(1.45191\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :1/2),\ 0.986 - 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42778 + 0.117360i\)
\(L(\frac12)\) \(\approx\) \(1.42778 + 0.117360i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.445 + 1.34i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-2.55 - 2.11i)T \)
good5 \( 1 + (-1.48 - 2.04i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.07 - 3.30i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.89 + 1.37i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.12 + 2.92i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-6.47 - 2.10i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.17iT - 23T^{2} \)
29 \( 1 + (0.252 + 0.776i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.49 - 4.81i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.96 + 0.639i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (4.24 + 1.38i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.90iT - 43T^{2} \)
47 \( 1 + (6.31 + 2.05i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.16 + 11.2i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.47 + 7.60i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.77 - 5.65i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.83T + 67T^{2} \)
71 \( 1 + (-3.94 - 5.43i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.23 + 1.37i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.7 + 7.78i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.34 - 3.23i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 2.99T + 89T^{2} \)
97 \( 1 + (5.74 + 4.17i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.87348010908723500314630498193, −11.14243561134045752634862002621, −10.05301920589605705198400955435, −9.553599083022379422413807664935, −8.540157220203249521093285559498, −6.74072219152351089173131347014, −5.58822817257979533867916022896, −4.75081932143079923287258780180, −3.14615166175649897656643547272, −2.13995704377500371530677410054, 1.21055543781722145454577757462, 3.76454663796497101638962225121, 4.95560368173052477361073706865, 5.87668776765658736817883200270, 7.03817569201903122578266943835, 7.75500116451737186216670515048, 8.948426354934972825652873394882, 9.661995170702648675191620010417, 11.21660006798067344550376940010, 12.14051626502511488712180430484

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