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Properties

Label 2-264-88.19-c1-0-18
Degree $2$
Conductor $264$
Sign $0.997 + 0.0748i$
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 + (−0.309 − 0.951i)3-s + (1.44 + 1.38i)4-s + (0.509 − 0.701i)5-s + (0.0942 − 1.41i)6-s + (0.919 − 2.83i)7-s + (1.17 + 2.57i)8-s + (−0.809 + 0.587i)9-s + (1.03 − 0.653i)10-s + (−0.848 + 3.20i)11-s + (0.865 − 1.80i)12-s + (3.09 − 2.25i)13-s + (2.69 − 3.23i)14-s + (−0.825 − 0.268i)15-s + (0.189 + 3.99i)16-s + (1.71 − 2.36i)17-s + ⋯
L(s)  = 1  + (0.928 + 0.371i)2-s + (−0.178 − 0.549i)3-s + (0.723 + 0.690i)4-s + (0.228 − 0.313i)5-s + (0.0384 − 0.576i)6-s + (0.347 − 1.07i)7-s + (0.415 + 0.909i)8-s + (−0.269 + 0.195i)9-s + (0.328 − 0.206i)10-s + (−0.255 + 0.966i)11-s + (0.249 − 0.520i)12-s + (0.859 − 0.624i)13-s + (0.720 − 0.864i)14-s + (−0.213 − 0.0692i)15-s + (0.0473 + 0.998i)16-s + (0.415 − 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.997 + 0.0748i$
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: \((2,\ 264,\ (\ :1/2),\ 0.997 + 0.0748i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11351 - 0.0792143i\)
\(L(\frac12)\) \(\approx\) \(2.11351 - 0.0792143i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.848 - 3.20i)T \)
good5 \( 1 + (-0.509 + 0.701i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.919 + 2.83i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.09 + 2.25i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.71 + 2.36i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (7.06 - 2.29i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.40iT - 23T^{2} \)
29 \( 1 + (3.25 - 10.0i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.24 + 5.84i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.72 + 1.21i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.570 + 0.185i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.21iT - 43T^{2} \)
47 \( 1 + (2.32 - 0.754i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.889 + 1.22i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.80 - 8.64i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.09 + 4.42i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.01T + 67T^{2} \)
71 \( 1 + (4.16 - 5.73i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.44 - 1.44i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-13.4 + 9.78i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.62 - 4.99i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 9.87T + 89T^{2} \)
97 \( 1 + (-0.462 + 0.335i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.38123727852203648582066716075, −11.03952525769799764965340182531, −10.48534268682161872606027905942, −8.770765063908659178742950803134, −7.63618955924293421689889805183, −7.02848434699224389924373687994, −5.81921688155946082492966814155, −4.80234319927562974108043137506, −3.62023569432409689955396531214, −1.79284168361734077437135549270, 2.14608224483565972913182124890, 3.47652641769737825028492633391, 4.68271744204848851914876662484, 5.89867730675111771818814210315, 6.36441984194571477773825468896, 8.247429513956451565294480845162, 9.211473945111140315004468804654, 10.49321501870295531312870353002, 11.09434993555297389054604506675, 11.87172343146715409378118289143

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