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Properties

Label 1-648-648.349-r0-0-0
Degree $1$
Conductor $648$
Sign $0.249 + 0.968i$
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  + (0.835 + 0.549i)5-s + (−0.686 − 0.727i)7-s + (−0.893 − 0.448i)11-s + (−0.396 + 0.918i)13-s + (0.766 + 0.642i)17-s + (−0.766 + 0.642i)19-s + (−0.686 + 0.727i)23-s + (0.396 + 0.918i)25-s + (0.993 + 0.116i)29-s + (0.973 + 0.230i)31-s + (−0.173 − 0.984i)35-s + (−0.173 + 0.984i)37-s + (0.597 + 0.802i)41-s + (0.0581 + 0.998i)43-s + (0.973 − 0.230i)47-s + ⋯
L(s)  = 1  + (0.835 + 0.549i)5-s + (−0.686 − 0.727i)7-s + (−0.893 − 0.448i)11-s + (−0.396 + 0.918i)13-s + (0.766 + 0.642i)17-s + (−0.766 + 0.642i)19-s + (−0.686 + 0.727i)23-s + (0.396 + 0.918i)25-s + (0.993 + 0.116i)29-s + (0.973 + 0.230i)31-s + (−0.173 − 0.984i)35-s + (−0.173 + 0.984i)37-s + (0.597 + 0.802i)41-s + (0.0581 + 0.998i)43-s + (0.973 − 0.230i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.249 + 0.968i$
Analytic conductor: \(3.00929\)
Root analytic conductor: \(3.00929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ 0.249 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9496293324 + 0.7360182719i\)
\(L(\frac12)\) \(\approx\) \(0.9496293324 + 0.7360182719i\)
\(L(1)\) \(\approx\) \(1.010739269 + 0.2097401004i\)
\(L(1)\) \(\approx\) \(1.010739269 + 0.2097401004i\)

Euler product

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

−22.63050499603826586116079261651, −21.84909508187568104381112561702, −21.054868754411305731034220238114, −20.38359671481709263859340713444, −19.42593206084965208946125403662, −18.487176154825719407209065757950, −17.76410752559957393393241397385, −17.01169995502979079232070527320, −15.93442047422672253609074277749, −15.448158255754102487376465865589, −14.280986343257798693747704588107, −13.39342830006274626075565878303, −12.53190818974094325529839719048, −12.17919297544392788202347241712, −10.50926236079335631563715526010, −10.02946875970040065454012353550, −9.08836793319404867399510201444, −8.26710353954525731572293001922, −7.14817516467452097863634337937, −5.97788382546837506554172563372, −5.395962079194871019218458851956, −4.41749337659815047938570859724, −2.75950291880561448339018326911, −2.33522866191566628939046261617, −0.59676721269880252333403854953, 1.37885285898213144622582258884, 2.57775554310531459436608734394, 3.48484906905493643161224171612, 4.63495878217119524110860269468, 5.91343270147358219331895613140, 6.46997595581612506553186929380, 7.501382414209702219041886743203, 8.47262687884787414975249613054, 9.86061342760744939708727866699, 10.09561084076114760116432672849, 11.040738918521177107168194698, 12.22443838319555744858210844252, 13.1607913916762880957983619177, 13.87428554121801639199612308703, 14.50785975732482702494236068241, 15.65025050851040746097845352910, 16.5798538483414593711970106840, 17.17156251520498344492694358891, 18.1349640092519622418775114865, 19.03456914769113055148753804891, 19.52769649652002067039266133737, 20.83404504048794184019368440880, 21.41681880429902439462145841795, 22.07735453180307901996298392554, 23.25737872989449918207252814670

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