Properties

Degree 1
Conductor $ 2^{6} $
Sign $-0.634 + 0.773i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(64\)    =    \(2^{6}\)
\( \varepsilon \)  =  $-0.634 + 0.773i$
motivic weight  =  \(0\)
character  :  $\chi_{64} (45, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 64,\ (0:\ ),\ -0.634 + 0.773i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.2341664904 + 0.4951034460i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.2341664904 + 0.4951034460i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5842022621 + 0.3719709757i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5842022621 + 0.3719709757i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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