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Properties

Label 1-2e6-64.11-r1-0-0
Degree $1$
Conductor $64$
Sign $0.0980 - 0.995i$
Analytic cond. $6.87775$
Root an. cond. $6.87775$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 + 0.382i)3-s + (−0.382 + 0.923i)5-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s + (−0.382 − 0.923i)13-s − i·15-s − i·17-s + (−0.382 − 0.923i)19-s + (0.923 + 0.382i)21-s + (0.707 − 0.707i)23-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + (−0.923 + 0.382i)29-s + 31-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)3-s + (−0.382 + 0.923i)5-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s + (−0.382 − 0.923i)13-s − i·15-s − i·17-s + (−0.382 − 0.923i)19-s + (0.923 + 0.382i)21-s + (0.707 − 0.707i)23-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + (−0.923 + 0.382i)29-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.0980 - 0.995i$
Analytic conductor: \(6.87775\)
Root analytic conductor: \(6.87775\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 64,\ (1:\ ),\ 0.0980 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4505741675 - 0.4083766211i\)
\(L(\frac12)\) \(\approx\) \(0.4505741675 - 0.4083766211i\)
\(L(1)\) \(\approx\) \(0.6534140122 + 0.02823038082i\)
\(L(1)\) \(\approx\) \(0.6534140122 + 0.02823038082i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-0.923 + 0.382i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
13 \( 1 + (-0.382 - 0.923i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.382 - 0.923i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.923 + 0.382i)T \)
31 \( 1 + T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (-0.923 - 0.382i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.923 - 0.382i)T \)
59 \( 1 + (0.382 - 0.923i)T \)
61 \( 1 + (0.923 - 0.382i)T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.382 - 0.923i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−32.237074500346725376716797353730, −31.25384372408038803104778004421, −29.82951971613480346793814560593, −28.79833155114589331719916481048, −28.109712920684302158917702211028, −27.046798273974789547759064756991, −25.277885316446856328028497071713, −24.380330876723833217063579473402, −23.42479695061463383845106098694, −22.23555497360364543160344544080, −21.26506908030573864391837054139, −19.46561306488694518317905094765, −18.86393153401818673907991135603, −17.05789504462062615819538747621, −16.591977302773788730452216821624, −15.23042199544198857971656979907, −13.33666682970962698972751609955, −12.258581575096350863810712814718, −11.550509152836069582375818816257, −9.731024079720704359368906896763, −8.41707953385030083083148488547, −6.69259275560829598569791420159, −5.57086557765859406277045394638, −4.04497410171680012159923388712, −1.51703771923268846367827641883, 0.37696370110948142502062837082, 3.21822909877469404263418242972, 4.63341333263358654232006080865, 6.44348141345438716738717527423, 7.24198445314439516609217076659, 9.5435846936499721662189272917, 10.58611192808475354471122767590, 11.60390903867584362499412235579, 12.94911184117007829335898505629, 14.62747647495937132907473064783, 15.70200600170526980813803087608, 16.91477622057863113389442684927, 17.90877089063807807681606523881, 19.249047977531212453659012485649, 20.42518899557517560989062754636, 22.17808976075950546942528075404, 22.612106817389432657579450307253, 23.54317732402161448156443787738, 25.13262762853263714187224999535, 26.55613338202532599406241381129, 27.21857783412603254920916688577, 28.393905520515572365843552326576, 29.7560920181690075197737698358, 30.18293295079110515675817196123, 31.98169953321532619959745591125

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