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Properties

Label 2-264-88.51-c1-0-11
Degree $2$
Conductor $264$
Sign $0.688 - 0.724i$
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 + 0.137i)2-s + (−0.309 + 0.951i)3-s + (1.96 + 0.387i)4-s + (2.02 + 2.78i)5-s + (−0.565 + 1.29i)6-s + (−1.45 − 4.47i)7-s + (2.70 + 0.815i)8-s + (−0.809 − 0.587i)9-s + (2.46 + 4.20i)10-s + (−2.16 + 2.51i)11-s + (−0.974 + 1.74i)12-s + (−2.66 − 1.93i)13-s + (−1.42 − 6.49i)14-s + (−3.27 + 1.06i)15-s + (3.69 + 1.52i)16-s + (1.49 + 2.05i)17-s + ⋯
L(s)  = 1  + (0.995 + 0.0973i)2-s + (−0.178 + 0.549i)3-s + (0.981 + 0.193i)4-s + (0.905 + 1.24i)5-s + (−0.231 + 0.529i)6-s + (−0.549 − 1.68i)7-s + (0.957 + 0.288i)8-s + (−0.269 − 0.195i)9-s + (0.780 + 1.32i)10-s + (−0.653 + 0.756i)11-s + (−0.281 + 0.504i)12-s + (−0.738 − 0.536i)13-s + (−0.381 − 1.73i)14-s + (−0.846 + 0.274i)15-s + (0.924 + 0.380i)16-s + (0.362 + 0.499i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.688 - 0.724i$
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: \((2,\ 264,\ (\ :1/2),\ 0.688 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06123 + 0.884614i\)
\(L(\frac12)\) \(\approx\) \(2.06123 + 0.884614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.137i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (2.16 - 2.51i)T \)
good5 \( 1 + (-2.02 - 2.78i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.45 + 4.47i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.66 + 1.93i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.49 - 2.05i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.674 - 0.219i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.35iT - 23T^{2} \)
29 \( 1 + (1.09 + 3.38i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.02 - 2.78i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-4.09 + 1.33i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (6.14 + 1.99i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.37iT - 43T^{2} \)
47 \( 1 + (-2.10 - 0.682i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.32 - 5.95i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.95 - 6.01i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.82 + 1.32i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + (0.411 + 0.566i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-9.39 + 3.05i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.44 - 1.77i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.890 - 1.22i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.52T + 89T^{2} \)
97 \( 1 + (-10.6 - 7.77i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.27827932629761816254711528296, −10.69112371939645558037335937698, −10.49805526004964896375760612806, −9.866917119115437894996615244463, −7.64705592749110340212974055932, −6.93413230006872842235544674812, −6.05701027442820251146714632439, −4.78994551925637957725850012546, −3.63646960532935427424614164873, −2.51122036070969813380489543251, 1.81690737006888590824220097196, 2.97230303246797989866802116083, 5.12851218536112792880866398745, 5.48477772476774232988498309098, 6.37982110922044307979611894652, 7.85380362664563448890775861624, 9.088182950450574343096564757450, 9.792801527388423640959556074168, 11.46622202678754175993056063178, 12.03872638046641879516335079206

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