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Properties

Label 2-264-88.51-c1-0-14
Degree $2$
Conductor $264$
Sign $-0.491 + 0.870i$
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  + (−1.40 + 0.167i)2-s + (−0.309 + 0.951i)3-s + (1.94 − 0.471i)4-s + (−1.03 − 1.43i)5-s + (0.274 − 1.38i)6-s + (−0.430 − 1.32i)7-s + (−2.65 + 0.988i)8-s + (−0.809 − 0.587i)9-s + (1.70 + 1.83i)10-s + (−2.97 + 1.45i)11-s + (−0.152 + 1.99i)12-s + (−2.66 − 1.93i)13-s + (0.827 + 1.78i)14-s + (1.68 − 0.546i)15-s + (3.55 − 1.83i)16-s + (−2.21 − 3.05i)17-s + ⋯
L(s)  = 1  + (−0.992 + 0.118i)2-s + (−0.178 + 0.549i)3-s + (0.971 − 0.235i)4-s + (−0.464 − 0.639i)5-s + (0.111 − 0.566i)6-s + (−0.162 − 0.501i)7-s + (−0.936 + 0.349i)8-s + (−0.269 − 0.195i)9-s + (0.537 + 0.580i)10-s + (−0.898 + 0.439i)11-s + (−0.0439 + 0.575i)12-s + (−0.738 − 0.536i)13-s + (0.221 + 0.478i)14-s + (0.434 − 0.141i)15-s + (0.888 − 0.458i)16-s + (−0.537 − 0.740i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $-0.491 + 0.870i$
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: \((2,\ 264,\ (\ :1/2),\ -0.491 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152124 - 0.260570i\)
\(L(\frac12)\) \(\approx\) \(0.152124 - 0.260570i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.167i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (2.97 - 1.45i)T \)
good5 \( 1 + (1.03 + 1.43i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.430 + 1.32i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.66 + 1.93i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.21 + 3.05i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.399 + 0.129i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.44iT - 23T^{2} \)
29 \( 1 + (2.45 + 7.54i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.816 + 1.12i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (8.91 - 2.89i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.29 - 1.06i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.60iT - 43T^{2} \)
47 \( 1 + (0.727 + 0.236i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.54 + 3.50i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.69 - 11.3i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (8.45 - 6.14i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.95T + 67T^{2} \)
71 \( 1 + (-7.57 - 10.4i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (9.14 - 2.97i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.69 + 3.41i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.86 + 12.2i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + (11.2 + 8.14i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.48366791419194479157629471381, −10.38587117604143172502757582264, −9.878007942088722700492325184012, −8.773260041818186146775386442514, −7.86419743563098861558370997261, −6.98384610412197938435324224721, −5.52220598541029104496977145557, −4.40530318528631985517910838775, −2.63808800402877671827281068397, −0.30328539145895062498561120081, 2.08656000579286150332395777641, 3.32682275094340826310450469354, 5.46630805833872679683698751033, 6.70026139726618308405666918712, 7.40173548848041209253618441895, 8.392483071627123604165243524849, 9.300212456750052114552435648741, 10.61930403723312885757439257133, 11.07548296995515149974046106630, 12.15980382861379398264439648719

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