Properties

Label 1-648-648.133-r0-0-0
Degree $1$
Conductor $648$
Sign $0.713 - 0.700i$
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  + (0.0581 − 0.998i)5-s + (0.973 − 0.230i)7-s + (0.835 − 0.549i)11-s + (0.993 − 0.116i)13-s + (0.766 + 0.642i)17-s + (−0.766 + 0.642i)19-s + (0.973 + 0.230i)23-s + (−0.993 − 0.116i)25-s + (−0.597 + 0.802i)29-s + (−0.286 − 0.957i)31-s + (−0.173 − 0.984i)35-s + (−0.173 + 0.984i)37-s + (0.396 − 0.918i)41-s + (−0.893 − 0.448i)43-s + (−0.286 + 0.957i)47-s + ⋯
L(s)  = 1  + (0.0581 − 0.998i)5-s + (0.973 − 0.230i)7-s + (0.835 − 0.549i)11-s + (0.993 − 0.116i)13-s + (0.766 + 0.642i)17-s + (−0.766 + 0.642i)19-s + (0.973 + 0.230i)23-s + (−0.993 − 0.116i)25-s + (−0.597 + 0.802i)29-s + (−0.286 − 0.957i)31-s + (−0.173 − 0.984i)35-s + (−0.173 + 0.984i)37-s + (0.396 − 0.918i)41-s + (−0.893 − 0.448i)43-s + (−0.286 + 0.957i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(3.00929\)
Root analytic conductor: \(3.00929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.643561013 - 0.6714685617i\)
\(L(\frac12)\) \(\approx\) \(1.643561013 - 0.6714685617i\)
\(L(1)\) \(\approx\) \(1.278589440 - 0.2750018610i\)
\(L(1)\) \(\approx\) \(1.278589440 - 0.2750018610i\)

Euler product

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

−23.181937332588264677330237037554, −22.10827562608724344905693029661, −21.33786595765483861338552954816, −20.68811926542966435927010708184, −19.586780302982333515171256972606, −18.77069340007475279063159079173, −18.042395661674644700344195632898, −17.40698732979295836956825624621, −16.38468542062911060674781853473, −15.214363216451300759484608840452, −14.708913343442456897299651267777, −14.00302698969540413255229872829, −12.99065939799367985805920485736, −11.74242784511662828183718796789, −11.25617801311402996288504858042, −10.4084028913712848362533977123, −9.319223831175649468345878406768, −8.453258369993519471138544678453, −7.35808069600010820575444751576, −6.65831777902845973459070771430, −5.621576168160667304381789613916, −4.51350297771317369454844191017, −3.50968600944323792176649637263, −2.38085780988113052509318288699, −1.34177515439063274395900201134, 1.106466147540304748168569438513, 1.71837392342895109331278009446, 3.532315431134113887574654881073, 4.25717184883867366911931535749, 5.37047607868132075938986144889, 6.084194802366596580689668787664, 7.4151000118968441709589083047, 8.50099404062169371794158165305, 8.75637581951735345124515745926, 10.05843378754295832374530108640, 11.073791175111950587769005869519, 11.76050833541656967074342991322, 12.77319955329200819357980213230, 13.51548838268534318809458990089, 14.47009230652777427982886724551, 15.18720122630594413726522501353, 16.477367249384816277965495625856, 16.8551074702213950470220622598, 17.665809590009526142110524220851, 18.745965466731872884275877805275, 19.47882181166217118720299451516, 20.67634336522742463346537008974, 20.858176482021362281436391958114, 21.773826885436123893164878260024, 22.88288502259093684611152827393

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