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Properties

Label 2-54e2-27.23-c0-0-0
Degree $2$
Conductor $2916$
Sign $0.727 - 0.686i$
Analytic cond. $1.45527$
Root an. cond. $1.20634$
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  + (−0.766 + 0.642i)7-s + (0.939 − 0.342i)13-s + (0.5 + 0.866i)19-s + (−0.939 − 0.342i)25-s + (1.53 + 1.28i)31-s + (0.5 − 0.866i)37-s + (0.347 + 1.96i)43-s + (−0.766 + 0.642i)61-s + (0.939 − 0.342i)67-s + (0.5 + 0.866i)73-s + (0.939 + 0.342i)79-s + (−0.5 + 0.866i)91-s + (−0.173 − 0.984i)97-s + (−0.173 + 0.984i)103-s + 2·109-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)7-s + (0.939 − 0.342i)13-s + (0.5 + 0.866i)19-s + (−0.939 − 0.342i)25-s + (1.53 + 1.28i)31-s + (0.5 − 0.866i)37-s + (0.347 + 1.96i)43-s + (−0.766 + 0.642i)61-s + (0.939 − 0.342i)67-s + (0.5 + 0.866i)73-s + (0.939 + 0.342i)79-s + (−0.5 + 0.866i)91-s + (−0.173 − 0.984i)97-s + (−0.173 + 0.984i)103-s + 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $0.727 - 0.686i$
Analytic conductor: \(1.45527\)
Root analytic conductor: \(1.20634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2916} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :0),\ 0.727 - 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.143669494\)
\(L(\frac12)\) \(\approx\) \(1.143669494\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
11 \( 1 + (0.939 - 0.342i)T^{2} \)
13 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.766 - 0.642i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.068599850453655791651659899275, −8.241123853463255661706440961609, −7.70362848608231717513371549382, −6.50719878343694279860295492550, −6.10245197242967492859529917870, −5.33410021459310691741739930197, −4.24286962462135214257260730499, −3.36171007752057881481046274668, −2.60014746838821733989875094774, −1.26202861153496730637493813292, 0.826187442566461644081498312610, 2.23657376074035047292492061673, 3.35365478073951625119845304904, 3.99912509535050093962770833822, 4.91897650634200258814344574566, 5.98892774009193692682899479825, 6.53257500746484571793399642162, 7.31701934454965597380259191904, 8.068193278203258995740977338900, 8.912149198028223188285378394365

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