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Properties

Label 2-264-88.35-c1-0-20
Degree $2$
Conductor $264$
Sign $0.543 + 0.839i$
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 − 0.525i)2-s + (0.809 − 0.587i)3-s + (1.44 − 1.37i)4-s + (−0.945 + 0.307i)5-s + (0.753 − 1.19i)6-s + (−1.02 − 0.742i)7-s + (1.17 − 2.57i)8-s + (0.309 − 0.951i)9-s + (−1.08 + 0.899i)10-s + (2.83 + 1.71i)11-s + (0.361 − 1.96i)12-s + (0.118 − 0.363i)13-s + (−1.73 − 0.438i)14-s + (−0.584 + 0.804i)15-s + (0.196 − 3.99i)16-s + (−2.87 + 0.932i)17-s + ⋯
L(s)  = 1  + (0.928 − 0.371i)2-s + (0.467 − 0.339i)3-s + (0.724 − 0.689i)4-s + (−0.422 + 0.137i)5-s + (0.307 − 0.488i)6-s + (−0.386 − 0.280i)7-s + (0.416 − 0.909i)8-s + (0.103 − 0.317i)9-s + (−0.341 + 0.284i)10-s + (0.855 + 0.517i)11-s + (0.104 − 0.567i)12-s + (0.0327 − 0.100i)13-s + (−0.463 − 0.117i)14-s + (−0.150 + 0.207i)15-s + (0.0492 − 0.998i)16-s + (−0.696 + 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.543 + 0.839i$
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: \((2,\ 264,\ (\ :1/2),\ 0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98400 - 1.07861i\)
\(L(\frac12)\) \(\approx\) \(1.98400 - 1.07861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 + 0.525i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-2.83 - 1.71i)T \)
good5 \( 1 + (0.945 - 0.307i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.02 + 0.742i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.118 + 0.363i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.87 - 0.932i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.491 - 0.676i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 9.25iT - 23T^{2} \)
29 \( 1 + (-2.76 - 2.00i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (4.33 + 1.40i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.20 + 1.66i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.83 + 6.65i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.32iT - 43T^{2} \)
47 \( 1 + (-5.10 - 7.02i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.88 - 1.58i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.26 - 3.82i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.29 + 10.1i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 1.94T + 67T^{2} \)
71 \( 1 + (10.6 - 3.47i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.24 + 1.71i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.517 + 1.59i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.38 + 0.449i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + (-2.73 + 8.41i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.90261687499648327089401526887, −11.19022844097363713279867608469, −9.991905202608260747223844739733, −9.108557532352722977097242805102, −7.53873005462575617481041810348, −6.86705440183840945108076043351, −5.67211424591379296373816861018, −4.15334627592890801400904727331, −3.36161454644742475163936532646, −1.73792670141848043232633929989, 2.55304832260115977084695477088, 3.80401709363061731001586680389, 4.67821829741297482961193936109, 6.10122040817036774522970207521, 6.96295619634236361753195257002, 8.281019485341155583547523139950, 8.938383533955321627767976798890, 10.34613441995068592871035251518, 11.45934368535474696546090623276, 12.17487080368742639978880154430

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