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Properties

Label 2-648-216.11-c1-0-12
Degree $2$
Conductor $648$
Sign $0.937 + 0.347i$
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  + (−0.844 − 1.13i)2-s + (−0.572 + 1.91i)4-s + (4.00 + 1.45i)5-s + (1.46 + 0.258i)7-s + (2.65 − 0.970i)8-s + (−1.73 − 5.77i)10-s + (−1.57 − 4.33i)11-s + (2.82 + 3.36i)13-s + (−0.946 − 1.88i)14-s + (−3.34 − 2.19i)16-s + (−2.43 + 1.40i)17-s + (0.516 − 0.895i)19-s + (−5.08 + 6.84i)20-s + (−3.58 + 5.45i)22-s + (0.345 + 1.95i)23-s + ⋯
L(s)  = 1  + (−0.597 − 0.801i)2-s + (−0.286 + 0.958i)4-s + (1.79 + 0.651i)5-s + (0.554 + 0.0977i)7-s + (0.939 − 0.343i)8-s + (−0.547 − 1.82i)10-s + (−0.475 − 1.30i)11-s + (0.784 + 0.934i)13-s + (−0.252 − 0.503i)14-s + (−0.836 − 0.548i)16-s + (−0.590 + 0.340i)17-s + (0.118 − 0.205i)19-s + (−1.13 + 1.52i)20-s + (−0.764 + 1.16i)22-s + (0.0720 + 0.408i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.937 + 0.347i$
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: \((2,\ 648,\ (\ :1/2),\ 0.937 + 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53125 - 0.274237i\)
\(L(\frac12)\) \(\approx\) \(1.53125 - 0.274237i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.844 + 1.13i)T \)
3 \( 1 \)
good5 \( 1 + (-4.00 - 1.45i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-1.46 - 0.258i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (1.57 + 4.33i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-2.82 - 3.36i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (2.43 - 1.40i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.516 + 0.895i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.345 - 1.95i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.783 + 0.657i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-6.04 + 1.06i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (5.15 - 2.97i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.77 + 3.30i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.28 - 0.830i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.806 + 4.57i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 3.14T + 53T^{2} \)
59 \( 1 + (2.77 - 7.62i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.25 - 0.397i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-7.24 + 6.07i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.21 + 3.84i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.12 + 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.19 - 2.62i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (3.23 - 3.85i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-3.14 - 1.81i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.20 - 1.89i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.59763363296592925035812280438, −9.729489867134824890964533472565, −8.909461536471334665833858299411, −8.302286687305133594127259438096, −6.89020875506035003039962937107, −6.09555439631409311545672280042, −5.02156473872036471855800808141, −3.49038965354241068385621621346, −2.40639404679652937110838445305, −1.47569432239130877125431109248, 1.27161355808952217955571919441, 2.29637902161820997948642505483, 4.68614605108009992874098919565, 5.24294532725612527280741065270, 6.12830131724464289527038433339, 6.99721751515007249979801154023, 8.164679167378040009698139510177, 8.803970486221494371311030435182, 9.802651636345958859569764950357, 10.15573814438907536923289478128

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