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Properties

Label 2-2e6-16.5-c1-0-0
Degree $2$
Conductor $64$
Sign $0.923 + 0.382i$
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 − i)3-s + (−1 − i)5-s + 2i·7-s + i·9-s + (−1 − i)11-s + (−1 + i)13-s − 2·15-s − 2·17-s + (−3 + 3i)19-s + (2 + 2i)21-s − 6i·23-s − 3i·25-s + (4 + 4i)27-s + (3 − 3i)29-s + 8·31-s + ⋯
L(s)  = 1  + (0.577 − 0.577i)3-s + (−0.447 − 0.447i)5-s + 0.755i·7-s + 0.333i·9-s + (−0.301 − 0.301i)11-s + (−0.277 + 0.277i)13-s − 0.516·15-s − 0.485·17-s + (−0.688 + 0.688i)19-s + (0.436 + 0.436i)21-s − 1.25i·23-s − 0.600i·25-s + (0.769 + 0.769i)27-s + (0.557 − 0.557i)29-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.511042\)
Root analytic conductor: \(0.714872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.943451 - 0.187664i\)
\(L(\frac12)\) \(\approx\) \(0.943451 - 0.187664i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-1 + i)T - 3iT^{2} \)
5 \( 1 + (1 + i)T + 5iT^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + (1 + i)T + 11iT^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + (3 - 3i)T - 19iT^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (5 + 5i)T + 43iT^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (5 + 5i)T + 53iT^{2} \)
59 \( 1 + (-3 - 3i)T + 59iT^{2} \)
61 \( 1 + (9 - 9i)T - 61iT^{2} \)
67 \( 1 + (-5 + 5i)T - 67iT^{2} \)
71 \( 1 - 10iT - 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-1 + i)T - 83iT^{2} \)
89 \( 1 - 4iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−14.80890398541489430227393698201, −13.71646814124117271064522680575, −12.68493327599984495986802133024, −11.79680949406565484562975769216, −10.29444745064215992476530998944, −8.623372325765915389171432951705, −8.094575571028396639247074109329, −6.44149975752932406076517563358, −4.66177529723384040571861019470, −2.46719287874121892924646954170, 3.19980240997241295571649335002, 4.56124503667803827231168171669, 6.71055742683799316053347084224, 7.943227555938203405118009237953, 9.337445096027797491465193578904, 10.38797986358525437670973004875, 11.48778254946706635368882741265, 12.95059353984951873190801474811, 14.08609491620803761202087545846, 15.15806687836610164216618137463

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