Properties

Label 2-2e6-64.21-c1-0-0
Degree $2$
Conductor $64$
Sign $0.623 - 0.781i$
Analytic cond. $0.511042$
Root an. cond. $0.714872$
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  + (−1.30 − 0.538i)2-s + (0.344 + 1.73i)3-s + (1.42 + 1.40i)4-s + (−2.21 + 1.48i)5-s + (0.481 − 2.45i)6-s + (2.90 + 1.20i)7-s + (−1.10 − 2.60i)8-s + (−0.109 + 0.0455i)9-s + (3.70 − 0.745i)10-s + (1.75 + 0.349i)11-s + (−1.94 + 2.94i)12-s + (−5.20 − 3.47i)13-s + (−3.15 − 3.13i)14-s + (−3.33 − 3.33i)15-s + (0.0367 + 3.99i)16-s + (0.895 − 0.895i)17-s + ⋯
L(s)  = 1  + (−0.924 − 0.380i)2-s + (0.198 + 1.00i)3-s + (0.710 + 0.703i)4-s + (−0.992 + 0.663i)5-s + (0.196 − 1.00i)6-s + (1.09 + 0.454i)7-s + (−0.389 − 0.921i)8-s + (−0.0366 + 0.0151i)9-s + (1.17 − 0.235i)10-s + (0.529 + 0.105i)11-s + (−0.562 + 0.850i)12-s + (−1.44 − 0.965i)13-s + (−0.842 − 0.838i)14-s + (−0.860 − 0.860i)15-s + (0.00918 + 0.999i)16-s + (0.217 − 0.217i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.623 - 0.781i$
Analytic conductor: \(0.511042\)
Root analytic conductor: \(0.714872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :1/2),\ 0.623 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.564406 + 0.271716i\)
\(L(\frac12)\) \(\approx\) \(0.564406 + 0.271716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 + 0.538i)T \)
good3 \( 1 + (-0.344 - 1.73i)T + (-2.77 + 1.14i)T^{2} \)
5 \( 1 + (2.21 - 1.48i)T + (1.91 - 4.61i)T^{2} \)
7 \( 1 + (-2.90 - 1.20i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.75 - 0.349i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (5.20 + 3.47i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 + (-0.895 + 0.895i)T - 17iT^{2} \)
19 \( 1 + (-2.36 + 3.53i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (0.709 + 1.71i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-7.36 + 1.46i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + 1.14iT - 31T^{2} \)
37 \( 1 + (1.36 + 2.03i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (3.08 + 7.44i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.56 - 7.88i)T + (-39.7 - 16.4i)T^{2} \)
47 \( 1 + (6.65 - 6.65i)T - 47iT^{2} \)
53 \( 1 + (0.674 + 0.134i)T + (48.9 + 20.2i)T^{2} \)
59 \( 1 + (2.59 - 1.73i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-0.360 - 1.81i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (2.13 + 10.7i)T + (-61.8 + 25.6i)T^{2} \)
71 \( 1 + (-1.97 - 0.819i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (13.1 - 5.45i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-0.102 - 0.102i)T + 79iT^{2} \)
83 \( 1 + (-5.13 + 7.68i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (4.64 - 11.2i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + 12.8iT - 97T^{2} \)
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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

−15.23244504301042538237488202985, −14.59994362494028648004212570239, −12.30185869193176103546560748564, −11.48893920686933838382454087367, −10.49527554858146732717680344963, −9.484355431808400725583954683607, −8.179209576119112301675043602043, −7.19498621559889027406007640547, −4.65534601877950690043590956301, −3.02034551984469177109834923401, 1.51853916164783775869851629519, 4.74427112013467779052583814483, 6.88922588111798106204640881348, 7.75605837146025082126453485026, 8.481174945325450378031506265157, 10.04362589597097469164938106576, 11.71703840450323184825819001210, 12.11411463814842069863514532908, 13.96585611032523596728923450394, 14.76247173200403532696172892685

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