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Properties

Label 2-264-88.19-c1-0-9
Degree $2$
Conductor $264$
Sign $0.928 - 0.370i$
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  + (−1.23 + 0.684i)2-s + (−0.309 − 0.951i)3-s + (1.06 − 1.69i)4-s + (−0.409 + 0.563i)5-s + (1.03 + 0.965i)6-s + (−0.510 + 1.57i)7-s + (−0.154 + 2.82i)8-s + (−0.809 + 0.587i)9-s + (0.120 − 0.977i)10-s + (2.08 + 2.58i)11-s + (−1.93 − 0.486i)12-s + (4.92 − 3.57i)13-s + (−0.443 − 2.29i)14-s + (0.662 + 0.215i)15-s + (−1.74 − 3.60i)16-s + (2.23 − 3.07i)17-s + ⋯
L(s)  = 1  + (−0.874 + 0.484i)2-s + (−0.178 − 0.549i)3-s + (0.531 − 0.847i)4-s + (−0.183 + 0.251i)5-s + (0.421 + 0.394i)6-s + (−0.192 + 0.593i)7-s + (−0.0545 + 0.998i)8-s + (−0.269 + 0.195i)9-s + (0.0381 − 0.309i)10-s + (0.628 + 0.778i)11-s + (−0.559 − 0.140i)12-s + (1.36 − 0.991i)13-s + (−0.118 − 0.612i)14-s + (0.170 + 0.0555i)15-s + (−0.435 − 0.900i)16-s + (0.541 − 0.745i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.928 - 0.370i$
Analytic conductor: \(2.10805\)
Root analytic conductor: \(1.45191\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :1/2),\ 0.928 - 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819054 + 0.157521i\)
\(L(\frac12)\) \(\approx\) \(0.819054 + 0.157521i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 - 0.684i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-2.08 - 2.58i)T \)
good5 \( 1 + (0.409 - 0.563i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.510 - 1.57i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.92 + 3.57i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.23 + 3.07i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-5.66 + 1.84i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.64iT - 23T^{2} \)
29 \( 1 + (-1.33 + 4.10i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.09 + 5.64i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.02 - 0.657i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.736 + 0.239i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.1iT - 43T^{2} \)
47 \( 1 + (5.29 - 1.72i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.57 + 3.54i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.78 + 11.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.239 - 0.174i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + (-1.14 + 1.56i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (6.90 + 2.24i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.76 - 2.01i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.90 - 2.62i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 6.63T + 89T^{2} \)
97 \( 1 + (3.48 - 2.53i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.55978565350312673645934430868, −11.32426092116759666666273396968, −9.790222441258491855820123287073, −9.225268040794541466940856813572, −7.901069684097776700296796362832, −7.32306174014754700355890110582, −6.13397384600209789706731493438, −5.34451541953277445943939327450, −3.13155817063571228743402982088, −1.32177870282027304935334540061, 1.15797002980278910296962231459, 3.36797870720603449987654365591, 4.18353769369132019417630501874, 6.05760183248650774071187814439, 7.05054620462381685647562549593, 8.542694838365066744022644882927, 8.865475921598723874269018496599, 10.20209999583844327720959120868, 10.75274066527777379174773008972, 11.69970454903436704967710222391

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