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Properties

Label 2-648-72.61-c1-0-11
Degree $2$
Conductor $648$
Sign $0.209 - 0.977i$
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  + (−1.09 + 0.895i)2-s + (0.395 − 1.96i)4-s + (−0.866 + 0.5i)5-s + (−0.5 + 0.866i)7-s + (1.32 + 2.49i)8-s + (0.499 − 1.32i)10-s + (2.59 + 1.5i)11-s + (4.58 − 2.64i)13-s + (−0.228 − 1.39i)14-s + (−3.68 − 1.55i)16-s − 5.29·17-s − 5.29i·19-s + (0.637 + 1.89i)20-s + (−4.18 + 0.685i)22-s + (2.64 + 4.58i)23-s + ⋯
L(s)  = 1  + (−0.773 + 0.633i)2-s + (0.197 − 0.980i)4-s + (−0.387 + 0.223i)5-s + (−0.188 + 0.327i)7-s + (0.467 + 0.883i)8-s + (0.158 − 0.418i)10-s + (0.783 + 0.452i)11-s + (1.27 − 0.733i)13-s + (−0.0610 − 0.373i)14-s + (−0.921 − 0.387i)16-s − 1.28·17-s − 1.21i·19-s + (0.142 + 0.423i)20-s + (−0.892 + 0.146i)22-s + (0.551 + 0.955i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.209 - 0.977i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.209 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.732373 + 0.591854i\)
\(L(\frac12)\) \(\approx\) \(0.732373 + 0.591854i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 - 0.895i)T \)
3 \( 1 \)
good5 \( 1 + (0.866 - 0.5i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.58 + 2.64i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 + 5.29iT - 19T^{2} \)
23 \( 1 + (-2.64 - 4.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.19 - 3i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.29iT - 37T^{2} \)
41 \( 1 + (2.64 + 4.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.16 - 5.29i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + (-3.46 + 2i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 - 3T + 73T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.06 + 3.5i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.89704312522904316383102482581, −9.600630825344312853953835613113, −8.936925128858101876182836413317, −8.269764460329668108317278002985, −7.08911165921733754182815678670, −6.58617189503577965006640889739, −5.52201569231358184521678901558, −4.38256923011874471299996721401, −2.91238376205532063690637598470, −1.23763618543520277559053537568, 0.794904761691422534358901544123, 2.25376630075413671866525897956, 3.82647498992848099545548303974, 4.22942312961620754110201103796, 6.22073278149752322295854447789, 6.83159904606308058625499130310, 8.176836982103752664149266230057, 8.597577231016288386722731545344, 9.473082496409031834726823942036, 10.42558432180426206705223349473

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