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Properties

Label 2-648-216.11-c1-0-26
Degree $2$
Conductor $648$
Sign $0.661 + 0.750i$
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  + (1.30 − 0.549i)2-s + (1.39 − 1.43i)4-s + (−0.437 − 0.159i)5-s + (3.46 + 0.610i)7-s + (1.03 − 2.63i)8-s + (−0.657 + 0.0329i)10-s + (−0.485 − 1.33i)11-s + (1.62 + 1.93i)13-s + (4.84 − 1.10i)14-s + (−0.103 − 3.99i)16-s + (0.667 − 0.385i)17-s + (−3.66 + 6.34i)19-s + (−0.838 + 0.404i)20-s + (−1.36 − 1.47i)22-s + (−1.25 − 7.12i)23-s + ⋯
L(s)  = 1  + (0.921 − 0.388i)2-s + (0.697 − 0.716i)4-s + (−0.195 − 0.0711i)5-s + (1.30 + 0.230i)7-s + (0.364 − 0.931i)8-s + (−0.207 + 0.0104i)10-s + (−0.146 − 0.402i)11-s + (0.449 + 0.535i)13-s + (1.29 − 0.295i)14-s + (−0.0258 − 0.999i)16-s + (0.161 − 0.0934i)17-s + (−0.839 + 1.45i)19-s + (−0.187 + 0.0903i)20-s + (−0.291 − 0.313i)22-s + (−0.261 − 1.48i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.661 + 0.750i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.661 + 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.57871 - 1.16458i\)
\(L(\frac12)\) \(\approx\) \(2.57871 - 1.16458i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 + 0.549i)T \)
3 \( 1 \)
good5 \( 1 + (0.437 + 0.159i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-3.46 - 0.610i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (0.485 + 1.33i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-1.62 - 1.93i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.667 + 0.385i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.66 - 6.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.25 + 7.12i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.15 - 2.64i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-7.80 + 1.37i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-2.79 + 1.61i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.74 + 2.07i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.99 - 0.726i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.82 - 10.3i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 8.73T + 53T^{2} \)
59 \( 1 + (0.818 - 2.24i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (3.02 + 0.533i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (4.04 - 3.39i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-5.70 - 9.88i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.96 + 5.12i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.94 - 10.6i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (7.41 - 8.83i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (8.59 + 4.96i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.95 - 0.710i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.71013146459872665705445344758, −9.890791959049326096636660352613, −8.449056937099206100765093118375, −7.974451709908206098527009397007, −6.53753945778762485814212811928, −5.83673780710227853581006107695, −4.65825715673359591426269191151, −4.08564653163981209326569626849, −2.60812636527541916186540433001, −1.44371551219942035457748722735, 1.78826909730474424373741354908, 3.16644947398222504803395123124, 4.39462672483546383858865250890, 5.01452112798460739957643444996, 6.07535678390799391534655972528, 7.13971157940540860710429097071, 7.922087404997480175254391746474, 8.528011589061478689107532332249, 9.966661802488306773457868346917, 11.09352976896425242362680210608

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