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Properties

Label 2-264-88.35-c1-0-4
Degree $2$
Conductor $264$
Sign $0.829 - 0.557i$
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  + (−1.41 + 0.0416i)2-s + (0.809 − 0.587i)3-s + (1.99 − 0.117i)4-s + (−2.29 + 0.744i)5-s + (−1.11 + 0.864i)6-s + (2.61 + 1.90i)7-s + (−2.81 + 0.249i)8-s + (0.309 − 0.951i)9-s + (3.20 − 1.14i)10-s + (1.66 + 2.86i)11-s + (1.54 − 1.26i)12-s + (−0.947 + 2.91i)13-s + (−3.78 − 2.58i)14-s + (−1.41 + 1.94i)15-s + (3.97 − 0.470i)16-s + (6.10 − 1.98i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0294i)2-s + (0.467 − 0.339i)3-s + (0.998 − 0.0588i)4-s + (−1.02 + 0.332i)5-s + (−0.456 + 0.352i)6-s + (0.989 + 0.719i)7-s + (−0.996 + 0.0882i)8-s + (0.103 − 0.317i)9-s + (1.01 − 0.362i)10-s + (0.501 + 0.865i)11-s + (0.446 − 0.366i)12-s + (−0.262 + 0.808i)13-s + (−1.01 − 0.689i)14-s + (−0.365 + 0.503i)15-s + (0.993 − 0.117i)16-s + (1.47 − 0.480i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.829 - 0.557i$
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: \((2,\ 264,\ (\ :1/2),\ 0.829 - 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.872776 + 0.266069i\)
\(L(\frac12)\) \(\approx\) \(0.872776 + 0.266069i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0416i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-1.66 - 2.86i)T \)
good5 \( 1 + (2.29 - 0.744i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.61 - 1.90i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.947 - 2.91i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-6.10 + 1.98i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.09 + 1.50i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 0.597iT - 23T^{2} \)
29 \( 1 + (-7.47 - 5.42i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.12 - 0.366i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.797 + 1.09i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.34 - 5.97i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.87iT - 43T^{2} \)
47 \( 1 + (5.56 + 7.65i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (7.66 + 2.48i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.09 + 4.42i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.95 + 12.1i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 5.40T + 67T^{2} \)
71 \( 1 + (4.12 - 1.34i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.84 - 6.66i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.77 + 14.6i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (12.7 - 4.15i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + (0.126 - 0.390i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.87457860594404892772853180888, −11.26936487433994237151799489284, −9.961824974080301946605936840739, −9.029479766491477810383780302416, −8.101016672429296028421655224301, −7.47361846888110235094696878396, −6.52976459112714597462725546076, −4.78623767981785422167930527152, −3.10581426233948795357494698251, −1.66802897416856836971920512816, 1.07432425529104819180393282769, 3.17778734214807986003116637713, 4.32945102640986952967953170783, 5.94620816652963474400617419788, 7.69116684697780413277559603819, 7.908093232918101091344010350389, 8.756999079191376969159745178190, 10.05426423881603323201205828078, 10.74617754480921359920005140615, 11.68520670522799117833510882920

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