Properties

Label 2-648-648.131-c1-0-58
Degree $2$
Conductor $648$
Sign $0.717 + 0.696i$
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  + (−0.432 − 1.34i)2-s + (1.40 + 1.00i)3-s + (−1.62 + 1.16i)4-s + (0.250 − 4.29i)5-s + (0.744 − 2.33i)6-s + (0.552 + 2.33i)7-s + (2.27 + 1.68i)8-s + (0.975 + 2.83i)9-s + (−5.89 + 1.52i)10-s + (2.53 + 3.85i)11-s + (−3.46 + 0.00611i)12-s + (0.549 + 4.70i)13-s + (2.89 − 1.75i)14-s + (4.67 − 5.80i)15-s + (1.28 − 3.78i)16-s + (3.27 − 3.89i)17-s + ⋯
L(s)  = 1  + (−0.305 − 0.952i)2-s + (0.813 + 0.580i)3-s + (−0.812 + 0.582i)4-s + (0.111 − 1.92i)5-s + (0.304 − 0.952i)6-s + (0.208 + 0.880i)7-s + (0.803 + 0.595i)8-s + (0.325 + 0.945i)9-s + (−1.86 + 0.481i)10-s + (0.763 + 1.16i)11-s + (−0.999 + 0.00176i)12-s + (0.152 + 1.30i)13-s + (0.774 − 0.468i)14-s + (1.20 − 1.49i)15-s + (0.321 − 0.946i)16-s + (0.793 − 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.717 + 0.696i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.717 + 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61388 - 0.655101i\)
\(L(\frac12)\) \(\approx\) \(1.61388 - 0.655101i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.432 + 1.34i)T \)
3 \( 1 + (-1.40 - 1.00i)T \)
good5 \( 1 + (-0.250 + 4.29i)T + (-4.96 - 0.580i)T^{2} \)
7 \( 1 + (-0.552 - 2.33i)T + (-6.25 + 3.14i)T^{2} \)
11 \( 1 + (-2.53 - 3.85i)T + (-4.35 + 10.1i)T^{2} \)
13 \( 1 + (-0.549 - 4.70i)T + (-12.6 + 2.99i)T^{2} \)
17 \( 1 + (-3.27 + 3.89i)T + (-2.95 - 16.7i)T^{2} \)
19 \( 1 + (-4.26 + 3.58i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (0.680 + 0.161i)T + (20.5 + 10.3i)T^{2} \)
29 \( 1 + (-3.10 + 4.17i)T + (-8.31 - 27.7i)T^{2} \)
31 \( 1 + (-0.869 + 0.260i)T + (25.9 - 17.0i)T^{2} \)
37 \( 1 + (6.31 + 1.11i)T + (34.7 + 12.6i)T^{2} \)
41 \( 1 + (5.71 + 2.46i)T + (28.1 + 29.8i)T^{2} \)
43 \( 1 + (-4.78 - 2.40i)T + (25.6 + 34.4i)T^{2} \)
47 \( 1 + (-2.01 + 6.72i)T + (-39.2 - 25.8i)T^{2} \)
53 \( 1 + (3.66 - 6.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.24 - 4.93i)T + (-23.3 - 54.1i)T^{2} \)
61 \( 1 + (-2.78 + 2.63i)T + (3.54 - 60.8i)T^{2} \)
67 \( 1 + (1.77 + 2.38i)T + (-19.2 + 64.1i)T^{2} \)
71 \( 1 + (-4.28 + 1.56i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-6.06 - 2.20i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (2.59 - 1.11i)T + (54.2 - 57.4i)T^{2} \)
83 \( 1 + (11.7 - 5.06i)T + (56.9 - 60.3i)T^{2} \)
89 \( 1 + (-0.471 + 1.29i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.0633 - 1.08i)T + (-96.3 + 11.2i)T^{2} \)
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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

−10.01199001991296446949475136251, −9.303852817390843343373774793088, −9.136533316008927218751915614271, −8.364859353811777715404432444407, −7.34812690077895212103676516917, −5.25770441061127451470429093172, −4.71403382778515347366085071682, −3.92272508876483081076515721186, −2.34638575273537750525703012041, −1.39950389577851656754552058854, 1.26239406102204355101822203621, 3.28908469519106610838796579146, 3.64242686203628933571189544218, 5.76647593619908273606340575887, 6.40280736986935516850525225952, 7.26100332299504651030595145369, 7.82769115017809495781691337615, 8.550236408435097889842309563394, 9.941764059635752913257091203261, 10.32738951684228381004526867579

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