Properties

Label 1-3e3-27.11-r1-0-0
Degree $1$
Conductor $27$
Sign $-0.973 + 0.230i$
Analytic cond. $2.90155$
Root an. cond. $2.90155$
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  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (−0.766 + 0.642i)11-s + (0.173 − 0.984i)13-s + (−0.173 + 0.984i)14-s + (0.766 − 0.642i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.939 + 0.342i)20-s + (0.766 + 0.642i)22-s + (0.939 − 0.342i)23-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (−0.766 + 0.642i)11-s + (0.173 − 0.984i)13-s + (−0.173 + 0.984i)14-s + (0.766 − 0.642i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.939 + 0.342i)20-s + (0.766 + 0.642i)22-s + (0.939 − 0.342i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-0.973 + 0.230i$
Analytic conductor: \(2.90155\)
Root analytic conductor: \(2.90155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 27,\ (1:\ ),\ -0.973 + 0.230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05921041265 - 0.5065774553i\)
\(L(\frac12)\) \(\approx\) \(-0.05921041265 - 0.5065774553i\)
\(L(1)\) \(\approx\) \(0.4465541098 - 0.4213020739i\)
\(L(1)\) \(\approx\) \(0.4465541098 - 0.4213020739i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (-0.766 + 0.642i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.173 - 0.984i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−38.14748585470791634191199758469, −36.65414954387836433063715957172, −35.324007812268473141980116406429, −34.6032551363486151966910720753, −33.37914373523119456545712606541, −31.89060273007651327608339253228, −31.120543154667379999215267918855, −29.161769214902919219806521372071, −27.68663959252193311963148303947, −26.42362686748262217914851674450, −25.63015970793499628409406678245, −23.89912739879143438241082614699, −23.027631564054719980470720924423, −21.68004100222232808790321897485, −19.19716449836050570360196533730, −18.64107535239525937157780119572, −16.64921086860621634899223254886, −15.65784272347344243117203224705, −14.37634047361713257153511885309, −12.71382006953854536147982547109, −10.606114965047337592231145829, −8.8489952580935580482103306558, −7.31391374102176597511236372497, −5.912213949868429947069378661919, −3.67587321190866865665264571594, 0.40515979689655816385167494069, 3.11869405203256950744374240630, 4.83919863193021852520397902, 7.67624332891158225658961573183, 9.30272849829234547207330912408, 10.74735226845145709356964508007, 12.40138710650043564277517233115, 13.26198261813935985602005333439, 15.534029608088021273197870062108, 17.06790930075876897197040109459, 18.69728937412319122803231942658, 19.92909377559328978111419403560, 20.77015092437506355351238214470, 22.60941689075382846974945049000, 23.44910100705191497046355503754, 25.535973708211900024866295605071, 26.94782156542666580240618103479, 28.09090131467346742700868665622, 29.10439627877956792165226807424, 30.45846737693771441945897715639, 31.696014406351162532665383735159, 32.584456174519239319836971505542, 34.7838087009068832282993933988, 35.89273433976354570983526116374, 36.71054444770414453315070241699

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