Properties

Label 2-2e6-16.5-c3-0-4
Degree $2$
Conductor $64$
Sign $-0.167 + 0.985i$
Analytic cond. $3.77612$
Root an. cond. $1.94322$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.27 − 3.27i)3-s + (−12.6 − 12.6i)5-s − 13.8i·7-s + 5.59i·9-s + (−1.54 − 1.54i)11-s + (32.7 − 32.7i)13-s − 82.7·15-s + 18.6·17-s + (86.4 − 86.4i)19-s + (−45.3 − 45.3i)21-s + 134. i·23-s + 194. i·25-s + (106. + 106. i)27-s + (−59.7 + 59.7i)29-s + 31.5·31-s + ⋯
L(s)  = 1  + (0.629 − 0.629i)3-s + (−1.13 − 1.13i)5-s − 0.749i·7-s + 0.207i·9-s + (−0.0424 − 0.0424i)11-s + (0.699 − 0.699i)13-s − 1.42·15-s + 0.266·17-s + (1.04 − 1.04i)19-s + (−0.471 − 0.471i)21-s + 1.21i·23-s + 1.55i·25-s + (0.760 + 0.760i)27-s + (−0.382 + 0.382i)29-s + 0.182·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.167 + 0.985i$
Analytic conductor: \(3.77612\)
Root analytic conductor: \(1.94322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :3/2),\ -0.167 + 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.902146 - 1.06883i\)
\(L(\frac12)\) \(\approx\) \(0.902146 - 1.06883i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-3.27 + 3.27i)T - 27iT^{2} \)
5 \( 1 + (12.6 + 12.6i)T + 125iT^{2} \)
7 \( 1 + 13.8iT - 343T^{2} \)
11 \( 1 + (1.54 + 1.54i)T + 1.33e3iT^{2} \)
13 \( 1 + (-32.7 + 32.7i)T - 2.19e3iT^{2} \)
17 \( 1 - 18.6T + 4.91e3T^{2} \)
19 \( 1 + (-86.4 + 86.4i)T - 6.85e3iT^{2} \)
23 \( 1 - 134. iT - 1.21e4T^{2} \)
29 \( 1 + (59.7 - 59.7i)T - 2.43e4iT^{2} \)
31 \( 1 - 31.5T + 2.97e4T^{2} \)
37 \( 1 + (-89.1 - 89.1i)T + 5.06e4iT^{2} \)
41 \( 1 + 210. iT - 6.89e4T^{2} \)
43 \( 1 + (119. + 119. i)T + 7.95e4iT^{2} \)
47 \( 1 - 182.T + 1.03e5T^{2} \)
53 \( 1 + (26.1 + 26.1i)T + 1.48e5iT^{2} \)
59 \( 1 + (441. + 441. i)T + 2.05e5iT^{2} \)
61 \( 1 + (174. - 174. i)T - 2.26e5iT^{2} \)
67 \( 1 + (91.7 - 91.7i)T - 3.00e5iT^{2} \)
71 \( 1 + 348. iT - 3.57e5T^{2} \)
73 \( 1 + 299. iT - 3.89e5T^{2} \)
79 \( 1 - 943.T + 4.93e5T^{2} \)
83 \( 1 + (313. - 313. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.41e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.51e3T + 9.12e5T^{2} \)
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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

−13.75278204691826082885822742695, −13.18979160537757419926045199101, −12.05027251258514323618281047279, −10.87925272843315682559087466588, −9.123115032198351272194554372840, −8.009631817971933815054370164289, −7.35745855625804172472271602582, −5.09365977207373025911292522983, −3.52383567185329220876673831116, −0.993781933026243060606082500064, 2.99285370042561558265437297371, 4.08608695651972294645495602769, 6.27636006589347339299812053509, 7.73989611752257137067065004240, 8.888645356989188218768884474093, 10.14476444848607204978828419390, 11.38906105009746692706596602011, 12.23628056707127997570096392037, 14.09906715063862495820388102122, 14.82383038589923523958137698387

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