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Properties

Label 1-648-648.131-r0-0-0
Degree $1$
Conductor $648$
Sign $0.700 - 0.713i$
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  + (−0.993 − 0.116i)5-s + (−0.893 + 0.448i)7-s + (−0.396 + 0.918i)11-s + (−0.973 + 0.230i)13-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)19-s + (0.893 + 0.448i)23-s + (0.973 + 0.230i)25-s + (−0.286 − 0.957i)29-s + (0.835 − 0.549i)31-s + (0.939 − 0.342i)35-s + (0.939 + 0.342i)37-s + (0.686 + 0.727i)41-s + (0.597 + 0.802i)43-s + (−0.835 − 0.549i)47-s + ⋯
L(s)  = 1  + (−0.993 − 0.116i)5-s + (−0.893 + 0.448i)7-s + (−0.396 + 0.918i)11-s + (−0.973 + 0.230i)13-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)19-s + (0.893 + 0.448i)23-s + (0.973 + 0.230i)25-s + (−0.286 − 0.957i)29-s + (0.835 − 0.549i)31-s + (0.939 − 0.342i)35-s + (0.939 + 0.342i)37-s + (0.686 + 0.727i)41-s + (0.597 + 0.802i)43-s + (−0.835 − 0.549i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.700 - 0.713i$
Analytic conductor: \(3.00929\)
Root analytic conductor: \(3.00929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ 0.700 - 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6757659204 - 0.2837574765i\)
\(L(\frac12)\) \(\approx\) \(0.6757659204 - 0.2837574765i\)
\(L(1)\) \(\approx\) \(0.7406401160 + 0.02902675144i\)
\(L(1)\) \(\approx\) \(0.7406401160 + 0.02902675144i\)

Euler product

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

−22.90864217553222397509512403671, −22.36089012709534774971458241152, −21.37014681206617766340260801808, −20.38987640729027785506066600162, −19.39933572401654999604252354675, −19.2448790646725517171481394486, −18.18764703563630968763939409463, −16.8980855359064411088883038393, −16.438584814913465996518309117917, −15.5504614410919176059391631008, −14.74538647256007959372604107318, −13.82250064232365935900146177637, −12.65764003946793800107265133670, −12.31999119737768817042195052325, −10.93363637391382736144846126514, −10.51350554848279882734290355680, −9.33593605062512736898732743885, −8.30793436972052045893749168039, −7.5317943657347959586680203137, −6.64754361924728419613600419564, −5.62019728633519527512067018665, −4.39217555406543470792394496329, −3.48888421760937377113081484051, −2.72466842756228836248483071197, −0.894985581767590862976273930071, 0.48637777067409560080666140251, 2.401499841241995453672395513921, 3.11292057611477870826595072891, 4.46835729753930107933949220517, 5.05142973385840690578099318047, 6.51783487603398597886398210983, 7.27240063285888463576559217816, 8.04617890849642991889167794963, 9.43816224258116791076742671402, 9.665150150287290563200233959504, 11.188467031088953366845700921536, 11.80331746619508255357264912454, 12.70640182012124077263814315265, 13.325589109623237304275130936005, 14.71700195840979933251322538853, 15.38865896666998340505218631576, 15.96810352419586007852440953584, 16.909378410161198115626706641992, 17.86433477919866503866656717692, 18.85295194351232013863901703163, 19.50112072630070835583416517558, 20.129866103934431951374010592639, 21.0797129251652419197767951371, 22.15981096504550654570663183849, 22.81162760842440675002974795821

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