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Properties

Label 2-264-88.51-c1-0-1
Degree $2$
Conductor $264$
Sign $-0.959 - 0.281i$
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  + (0.274 + 1.38i)2-s + (−0.309 + 0.951i)3-s + (−1.84 + 0.760i)4-s + (1.03 + 1.43i)5-s + (−1.40 − 0.167i)6-s + (0.430 + 1.32i)7-s + (−1.56 − 2.35i)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 + (−1.72 + 0.961i)14-s + (−1.68 + 0.546i)15-s + (2.84 − 2.81i)16-s + (−2.21 − 3.05i)17-s + ⋯
L(s)  = 1  + (0.193 + 0.981i)2-s + (−0.178 + 0.549i)3-s + (−0.924 + 0.380i)4-s + (0.464 + 0.639i)5-s + (−0.573 − 0.0685i)6-s + (0.162 + 0.501i)7-s + (−0.552 − 0.833i)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 + (−0.460 + 0.256i)14-s + (−0.434 + 0.141i)15-s + (0.710 − 0.703i)16-s + (−0.537 − 0.740i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $-0.959 - 0.281i$
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: \((2,\ 264,\ (\ :1/2),\ -0.959 - 0.281i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.165473 + 1.15126i\)
\(L(\frac12)\) \(\approx\) \(0.165473 + 1.15126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.274 - 1.38i)T \)
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

−12.58458356061961734149803232203, −11.36502574384588544406086914989, −10.36672892225366797600084566083, −9.383583987327043585964400808459, −8.562238944882369151949140658497, −7.31403216567013205008075158033, −6.33648420683148327123730340534, −5.38823771603815011600757104560, −4.36704318570301974671182756265, −2.80594578507263664151203112678, 0.927456070732407382662824342853, 2.44401244053418153917761176697, 4.03707066901788072409470684649, 5.29817568629514564378586921685, 6.18006698531620900991898438165, 7.944094849478080293159501181980, 8.639423999706535837351241863454, 9.866880421423070860102994935613, 10.76298767783068760849663817605, 11.44745902992305133000550850948

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