Properties

Label 2-648-9.7-c1-0-7
Degree $2$
Conductor $648$
Sign $0.766 + 0.642i$
Analytic cond. $5.17430$
Root an. cond. $2.27471$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + (−1.5 + 2.59i)7-s + (2.5 − 4.33i)11-s + (−2 − 3.46i)13-s + 8·17-s + 2·19-s + (1 + 1.73i)23-s + (2 − 3.46i)25-s + (3 − 5.19i)29-s + (3.5 + 6.06i)31-s + 3·35-s − 6·37-s + (−3 − 5.19i)41-s + (1 − 1.73i)43-s + (3 − 5.19i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.566 + 0.981i)7-s + (0.753 − 1.30i)11-s + (−0.554 − 0.960i)13-s + 1.94·17-s + 0.458·19-s + (0.208 + 0.361i)23-s + (0.400 − 0.692i)25-s + (0.557 − 0.964i)29-s + (0.628 + 1.08i)31-s + 0.507·35-s − 0.986·37-s + (−0.468 − 0.811i)41-s + (0.152 − 0.264i)43-s + (0.437 − 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30546 - 0.475150i\)
\(L(\frac12)\) \(\approx\) \(1.30546 - 0.475150i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 8T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5T + 53T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.5 - 9.52i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.29425797916683476209565869922, −9.626758317022921323053723703404, −8.593886958864914171241138517523, −8.110546386813897342502550147185, −6.84029613150383195588173883153, −5.74709869924776261562529017044, −5.24767772441700568968181432170, −3.59906237316680168958216947870, −2.85537760843342155178755531408, −0.888267160271876403026464002014, 1.37186484275225425611946086557, 3.07476193194455228130293936967, 4.04682375699432952968016193007, 5.00692128552588531847269946150, 6.46245180557132342142837175121, 7.14958929160831739874800294386, 7.70722650355393378263148223792, 9.179920002163070702439689215692, 9.866151249978199832985838671133, 10.42583645999444522620363738423

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