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Properties

Label 1-2e6-64.61-r0-0-0
Degree $1$
Conductor $64$
Sign $0.995 - 0.0980i$
Analytic cond. $0.297214$
Root an. cond. $0.297214$
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

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

Invariants

Degree: \(1\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.995 - 0.0980i$
Analytic conductor: \(0.297214\)
Root analytic conductor: \(0.297214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 64,\ (0:\ ),\ 0.995 - 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8119408731 - 0.03988809729i\)
\(L(\frac12)\) \(\approx\) \(0.8119408731 - 0.03988809729i\)
\(L(1)\) \(\approx\) \(0.9068907647 - 0.03432106584i\)
\(L(1)\) \(\approx\) \(0.9068907647 - 0.03432106584i\)

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

−32.48925824230909645662251550873, −31.30283592035135928158501873299, −29.913437081431932851964587680133, −28.69991573711017823318648471161, −28.01222962343578063349895044473, −27.185941123978501335132454576794, −25.47246761480209979641954751498, −24.37485986558361741277942770280, −23.48278054620931789364482285476, −21.96304534460574951969261945297, −21.331499069182291517990092880287, −20.102153599416876182273319762010, −18.33640954462948472097894328018, −17.38285245533884050424775488682, −16.41900206014164688741025131006, −15.25891623490278612013348392065, −13.69032512900771916702788571802, −12.08110206505235517563023358434, −11.502895944557074895954851675882, −9.699973569552783693041862129061, −8.75333982704440655081199628750, −6.68440481280154351790128167086, −5.31292001496009521144944757391, −4.35027534755608764687885233959, −1.61395579689045687471854883394, 1.608912584339406317133466888601, 3.8867998504292872916667177647, 5.72560160837288939851634712019, 6.74129028435277390892562030394, 8.10007068692543716373527700145, 10.3495703628812869726471435203, 10.92086724369123080865205873707, 12.31656825464207934577479889254, 13.76001694195046695613877820987, 14.813705858993439915919031258048, 16.55618455407356057411478250208, 17.547230354954286438011313375996, 18.354273527506476576938099901225, 19.66696933749155809769867340566, 21.29169446961871940061828308820, 22.33989027011093792415485579118, 23.223854699107069746557901689172, 24.3399081087293895841568794329, 25.52318731369244143599038496171, 26.98816485460508106737450721647, 27.74346015813290854097809867185, 29.20043352005840924797988041409, 30.1324081960316766360203071220, 30.55132098673157590274788015355, 32.65874683918747615263502418071

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