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Properties

Label 1-2e6-64.45-r0-0-0
Degree $1$
Conductor $64$
Sign $-0.634 + 0.773i$
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  + (−0.382 + 0.923i)3-s + (−0.923 + 0.382i)5-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (0.382 + 0.923i)11-s + (−0.923 − 0.382i)13-s − i·15-s + i·17-s + (0.923 + 0.382i)19-s + (−0.382 − 0.923i)21-s + (0.707 + 0.707i)23-s + (0.707 − 0.707i)25-s + (0.923 − 0.382i)27-s + (0.382 − 0.923i)29-s − 31-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)3-s + (−0.923 + 0.382i)5-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (0.382 + 0.923i)11-s + (−0.923 − 0.382i)13-s − i·15-s + i·17-s + (0.923 + 0.382i)19-s + (−0.382 − 0.923i)21-s + (0.707 + 0.707i)23-s + (0.707 − 0.707i)25-s + (0.923 − 0.382i)27-s + (0.382 − 0.923i)29-s − 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.634 + 0.773i$
Analytic conductor: \(0.297214\)
Root analytic conductor: \(0.297214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 64,\ (0:\ ),\ -0.634 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2341664904 + 0.4951034460i\)
\(L(\frac12)\) \(\approx\) \(0.2341664904 + 0.4951034460i\)
\(L(1)\) \(\approx\) \(0.5842022621 + 0.3719709757i\)
\(L(1)\) \(\approx\) \(0.5842022621 + 0.3719709757i\)

Euler product

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

−31.72144930854946035552201926990, −30.7836971153937645317690942894, −29.475191800583939656942013535219, −28.93260564278486021192886719835, −27.42691436406110604466217727109, −26.49564125741052983808598722728, −24.86742112143333161999380821183, −24.07345807411019444349276347870, −23.0933775659905230123449843655, −22.13971418391187164835244023399, −20.14048749625888638445060855052, −19.477132799102398785148561852671, −18.41032275239441161713100439948, −16.798515080735922301398821911593, −16.234342515043831730147841556372, −14.309847255270968321699846702465, −13.13261752009893730337640382798, −12.024197091946205419071989604293, −11.03528588178380794474987440556, −9.13349255508956857971476327312, −7.60204815148777699537421810657, −6.73377127516193918498081014772, −4.952683446077630263402196970458, −3.14526182352538600857446704265, −0.71594056673812665595195479519, 3.034310578431362689139081590449, 4.34298369411387762035427196075, 5.86538571983263121771568525894, 7.454019811318934445290029604528, 9.18633024909061110072254772072, 10.23340222899596561796244948114, 11.64785193654906717994953256763, 12.52583010926954113200406553409, 14.793770404346961723949284124300, 15.339077648253987453558329581021, 16.49117866819529139747848140542, 17.77217828726911880197715840264, 19.2896014259617833649835152247, 20.189525426069310601616174718029, 21.75918118051078367921959175628, 22.53290014718844584362602461289, 23.387281034910836682577185459270, 25.05163842354468882405080850742, 26.25751485502270335600413955029, 27.23683668002495067212693489533, 28.11988244062868421873792127353, 29.112530174226472073815196532214, 30.68890043362054364962547502336, 31.67441060718899658019567904234, 32.60870090820936138779655596767

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