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Properties

Label 1-648-648.589-r0-0-0
Degree $1$
Conductor $648$
Sign $0.996 + 0.0774i$
Analytic cond. $3.00929$
Root an. cond. $3.00929$
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  + (−0.973 + 0.230i)5-s + (0.597 + 0.802i)7-s + (0.686 − 0.727i)11-s + (−0.893 − 0.448i)13-s + (−0.939 − 0.342i)17-s + (0.939 − 0.342i)19-s + (0.597 − 0.802i)23-s + (0.893 − 0.448i)25-s + (0.835 + 0.549i)29-s + (0.396 + 0.918i)31-s + (−0.766 − 0.642i)35-s + (−0.766 + 0.642i)37-s + (−0.0581 − 0.998i)41-s + (0.286 + 0.957i)43-s + (0.396 − 0.918i)47-s + ⋯
L(s)  = 1  + (−0.973 + 0.230i)5-s + (0.597 + 0.802i)7-s + (0.686 − 0.727i)11-s + (−0.893 − 0.448i)13-s + (−0.939 − 0.342i)17-s + (0.939 − 0.342i)19-s + (0.597 − 0.802i)23-s + (0.893 − 0.448i)25-s + (0.835 + 0.549i)29-s + (0.396 + 0.918i)31-s + (−0.766 − 0.642i)35-s + (−0.766 + 0.642i)37-s + (−0.0581 − 0.998i)41-s + (0.286 + 0.957i)43-s + (0.396 − 0.918i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.996 + 0.0774i$
Analytic conductor: \(3.00929\)
Root analytic conductor: \(3.00929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ 0.996 + 0.0774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.231344796 + 0.04778178578i\)
\(L(\frac12)\) \(\approx\) \(1.231344796 + 0.04778178578i\)
\(L(1)\) \(\approx\) \(0.9972464879 + 0.04022136744i\)
\(L(1)\) \(\approx\) \(0.9972464879 + 0.04022136744i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.973 + 0.230i)T \)
7 \( 1 + (0.597 + 0.802i)T \)
11 \( 1 + (0.686 - 0.727i)T \)
13 \( 1 + (-0.893 - 0.448i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.597 - 0.802i)T \)
29 \( 1 + (0.835 + 0.549i)T \)
31 \( 1 + (0.396 + 0.918i)T \)
37 \( 1 + (-0.766 + 0.642i)T \)
41 \( 1 + (-0.0581 - 0.998i)T \)
43 \( 1 + (0.286 + 0.957i)T \)
47 \( 1 + (0.396 - 0.918i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.686 + 0.727i)T \)
61 \( 1 + (0.993 - 0.116i)T \)
67 \( 1 + (0.835 - 0.549i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (-0.0581 + 0.998i)T \)
83 \( 1 + (0.0581 - 0.998i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (0.973 + 0.230i)T \)
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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

−22.873294084493110312191468934930, −22.23314913152938506939791541914, −21.06803270172462481203619656466, −20.26631373236914101069850119385, −19.674488205328651731996967443931, −19.00148778596024225484545127207, −17.61840886470565591865633870054, −17.24781440417879183453754399420, −16.239267761120707184343146546518, −15.33812390737678901910865878702, −14.6103709616303131124410508855, −13.765567678534670905286477617692, −12.69701265885332146218764324597, −11.75718042492321032864856855202, −11.30509462562788146328514839900, −10.11755717241289597133190587600, −9.23858209775895219824920854983, −8.15458287927224609443977749466, −7.3591893566448282084290438929, −6.74375054551593459792542314338, −5.12904646199308332962120340943, −4.36152046841708183512187354333, −3.67451949391875480404180910807, −2.15384658221268311236240061029, −0.927890121654728009194044866430, 0.87653415297628420869330170435, 2.48989037567229789027827588903, 3.28721418091192236007192753335, 4.56909561230774273210497652146, 5.25940772662134427052371239281, 6.604055791034379881676335655357, 7.35583510078404502982328873272, 8.512855783789000067795641713696, 8.908303012319222066205364900765, 10.307947563167459780057798227102, 11.26112060568160531322199939919, 11.8695825816921113915023055613, 12.560807811689891861089612089881, 13.9048330744985124933140892293, 14.618449495129771812828731767312, 15.45820853638776486398344464896, 16.04825080053966778012022801389, 17.168300344978285239129707771864, 18.02364737307744071148626679968, 18.814892512077076271968322610622, 19.63419501371914252394510929152, 20.22988339502558660251686465245, 21.36890535679069279372724642819, 22.2141089785437223279939202033, 22.652782567232191061773919521430

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