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Properties

Label 2-264-88.51-c1-0-17
Degree $2$
Conductor $264$
Sign $-0.954 + 0.298i$
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  + (0.0942 − 1.41i)2-s + (−0.309 + 0.951i)3-s + (−1.98 − 0.265i)4-s + (−0.509 − 0.701i)5-s + (1.31 + 0.525i)6-s + (−0.919 − 2.83i)7-s + (−0.562 + 2.77i)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 + (−4.08 + 1.03i)14-s + (0.825 − 0.268i)15-s + (3.85 + 1.05i)16-s + (1.71 + 2.36i)17-s + ⋯
L(s)  = 1  + (0.0666 − 0.997i)2-s + (−0.178 + 0.549i)3-s + (−0.991 − 0.132i)4-s + (−0.228 − 0.313i)5-s + (0.535 + 0.214i)6-s + (−0.347 − 1.07i)7-s + (−0.198 + 0.980i)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 + (−1.09 + 0.275i)14-s + (0.213 − 0.0692i)15-s + (0.964 + 0.263i)16-s + (0.415 + 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $-0.954 + 0.298i$
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: \((2,\ 264,\ (\ :1/2),\ -0.954 + 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105710 - 0.692135i\)
\(L(\frac12)\) \(\approx\) \(0.105710 - 0.692135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0942 + 1.41i)T \)
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

−11.30648394707611868649168986453, −10.54668184350795300132547186599, −10.05303958233939494550171898779, −8.800315946435241055032418019117, −7.973936193790375808983530160004, −6.30265835675553886043765682385, −4.91924916288537113121960492211, −4.05450093870946096273689261254, −2.87109925554258752187548371139, −0.53875678923354861604798846776, 2.49599198465523013506897806844, 4.36254434100484988345288965501, 5.51841124878731548833625431815, 6.54198912756083608086276229620, 7.31957787402754780650766854896, 8.317346070905669643999913684510, 9.334725346486356226278149422329, 10.24241270910897169804714414485, 11.92647014043004949012934783024, 12.34897175696101605050995537202

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