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Properties

Label 2-648-72.11-c1-0-25
Degree $2$
Conductor $648$
Sign $-0.173 + 0.984i$
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  + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s − 2.82i·8-s + (−2.44 − 1.41i)11-s + (−2.00 + 3.46i)16-s − 5.65i·17-s + 2·19-s + (1.99 + 3.46i)22-s + (2.5 − 4.33i)25-s + (4.89 − 2.82i)32-s + (−4.00 + 6.92i)34-s + (−2.44 − 1.41i)38-s + (9.79 − 5.65i)41-s + (5 − 8.66i)43-s − 5.65i·44-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (0.499 + 0.866i)4-s − 0.999i·8-s + (−0.738 − 0.426i)11-s + (−0.500 + 0.866i)16-s − 1.37i·17-s + 0.458·19-s + (0.426 + 0.738i)22-s + (0.5 − 0.866i)25-s + (0.866 − 0.499i)32-s + (−0.685 + 1.18i)34-s + (−0.397 − 0.229i)38-s + (1.53 − 0.883i)41-s + (0.762 − 1.32i)43-s − 0.852i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.511725 - 0.609850i\)
\(L(\frac12)\) \(\approx\) \(0.511725 - 0.609850i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.44 + 1.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.65iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-9.79 + 5.65i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (12.2 - 7.07i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.44 + 1.41i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.37332038098455669237591386821, −9.421008366686511328786793769446, −8.750820291661821421574738880850, −7.74598089995817991955549723890, −7.11666027261264462856554877039, −5.89701918867828366602938440360, −4.64679650058616018927627704801, −3.27485815069797705908059316950, −2.34556713359145431020197491604, −0.59878789389094428760888348338, 1.43649121692833088804410698127, 2.84349512476922018580285030845, 4.50783540317841986114746882219, 5.60893794420405578656809842938, 6.41393758956509253734278504980, 7.53021583784779964880884880040, 8.040722567071583597804193813531, 9.097612999576962826945590110729, 9.797927086663948667715671173817, 10.70580282581452824961779687407

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