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Properties

Label 2-264-88.35-c1-0-21
Degree $2$
Conductor $264$
Sign $0.993 - 0.110i$
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  + (1.31 + 0.515i)2-s + (0.809 − 0.587i)3-s + (1.46 + 1.35i)4-s + (2.47 − 0.803i)5-s + (1.36 − 0.357i)6-s + (−3.20 − 2.32i)7-s + (1.23 + 2.54i)8-s + (0.309 − 0.951i)9-s + (3.67 + 0.215i)10-s + (−3.14 − 1.04i)11-s + (1.98 + 0.233i)12-s + (−1.46 + 4.50i)13-s + (−3.02 − 4.71i)14-s + (1.52 − 2.10i)15-s + (0.318 + 3.98i)16-s + (−1.39 + 0.453i)17-s + ⋯
L(s)  = 1  + (0.931 + 0.364i)2-s + (0.467 − 0.339i)3-s + (0.734 + 0.678i)4-s + (1.10 − 0.359i)5-s + (0.558 − 0.145i)6-s + (−1.21 − 0.880i)7-s + (0.437 + 0.899i)8-s + (0.103 − 0.317i)9-s + (1.16 + 0.0681i)10-s + (−0.949 − 0.314i)11-s + (0.573 + 0.0675i)12-s + (−0.406 + 1.25i)13-s + (−0.808 − 1.26i)14-s + (0.394 − 0.543i)15-s + (0.0796 + 0.996i)16-s + (−0.338 + 0.110i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.993 - 0.110i$
Analytic conductor: \(2.10805\)
Root analytic conductor: \(1.45191\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :1/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.44479 + 0.135809i\)
\(L(\frac12)\) \(\approx\) \(2.44479 + 0.135809i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 - 0.515i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (3.14 + 1.04i)T \)
good5 \( 1 + (-2.47 + 0.803i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (3.20 + 2.32i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.46 - 4.50i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.39 - 0.453i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.733 + 1.00i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 5.10iT - 23T^{2} \)
29 \( 1 + (-0.392 - 0.285i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-7.31 - 2.37i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.66 + 7.79i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.57 - 2.16i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.05iT - 43T^{2} \)
47 \( 1 + (2.11 + 2.90i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (11.1 + 3.61i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (11.2 + 8.16i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.376 - 1.15i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + (-2.09 + 0.679i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.46 + 6.14i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.63 + 11.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (8.98 - 2.91i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + (-0.629 + 1.93i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.48008148176516323345510589869, −11.15465813769075542909751511702, −9.976244761216402824632159330847, −9.164538250437501917668259308779, −7.78655365868637954664280500916, −6.78580710116623128459153581288, −6.07009589470495075659926428438, −4.76633529813571482243457279734, −3.42674240463437091096489149160, −2.14579427016687170641817372436, 2.55921247226868910063816599912, 2.88585195415620940409552624740, 4.73152890581476657962153419680, 5.83699308421586957009743356438, 6.49355799416635682062590557172, 8.045121244945665489824348155326, 9.639052297294782058537734946572, 9.970040176592705477021283582162, 10.83570199408250277639351630810, 12.35734550950424575120935303681

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