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Properties

Label 2-663-663.332-c1-0-9
Degree $2$
Conductor $663$
Sign $0.989 + 0.147i$
Analytic cond. $5.29408$
Root an. cond. $2.30088$
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  + (−1.46 − 0.845i)2-s + (0.145 − 1.72i)3-s + (0.429 + 0.743i)4-s + (−0.342 − 0.827i)5-s + (−1.67 + 2.40i)6-s + (0.425 + 3.22i)7-s + 1.92i·8-s + (−2.95 − 0.501i)9-s + (−0.197 + 1.50i)10-s + (2.85 − 2.18i)11-s + (1.34 − 0.633i)12-s + (−2.89 + 2.14i)13-s + (2.10 − 5.08i)14-s + (−1.47 + 0.471i)15-s + (2.49 − 4.31i)16-s + (1.11 + 3.97i)17-s + ⋯
L(s)  = 1  + (−1.03 − 0.597i)2-s + (0.0838 − 0.996i)3-s + (0.214 + 0.371i)4-s + (−0.153 − 0.369i)5-s + (−0.682 + 0.981i)6-s + (0.160 + 1.22i)7-s + 0.682i·8-s + (−0.985 − 0.167i)9-s + (−0.0624 + 0.474i)10-s + (0.859 − 0.659i)11-s + (0.388 − 0.182i)12-s + (−0.803 + 0.595i)13-s + (0.563 − 1.36i)14-s + (−0.381 + 0.121i)15-s + (0.622 − 1.07i)16-s + (0.269 + 0.962i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $0.989 + 0.147i$
Analytic conductor: \(5.29408\)
Root analytic conductor: \(2.30088\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{663} (332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 663,\ (\ :1/2),\ 0.989 + 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.666756 - 0.0495545i\)
\(L(\frac12)\) \(\approx\) \(0.666756 - 0.0495545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.145 + 1.72i)T \)
13 \( 1 + (2.89 - 2.14i)T \)
17 \( 1 + (-1.11 - 3.97i)T \)
good2 \( 1 + (1.46 + 0.845i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.342 + 0.827i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.425 - 3.22i)T + (-6.76 + 1.81i)T^{2} \)
11 \( 1 + (-2.85 + 2.18i)T + (2.84 - 10.6i)T^{2} \)
19 \( 1 + (-2.62 - 4.55i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.21 - 9.21i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (5.67 - 4.35i)T + (7.50 - 28.0i)T^{2} \)
31 \( 1 + (-0.832 - 2.01i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-0.729 + 5.54i)T + (-35.7 - 9.57i)T^{2} \)
41 \( 1 + (-2.93 + 2.25i)T + (10.6 - 39.6i)T^{2} \)
43 \( 1 + (4.15 + 1.11i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-5.13 + 5.13i)T - 47iT^{2} \)
53 \( 1 + (-2.77 - 2.77i)T + 53iT^{2} \)
59 \( 1 + (0.620 - 0.358i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.26 - 5.55i)T + (-15.7 - 58.9i)T^{2} \)
67 \( 1 + (9.49 - 2.54i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.498 + 0.382i)T + (18.3 + 68.5i)T^{2} \)
73 \( 1 + (2.37 + 5.72i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-13.8 + 5.72i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 - 6.09iT - 83T^{2} \)
89 \( 1 + (-2.86 - 10.6i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.83 + 1.40i)T + (25.1 + 93.6i)T^{2} \)
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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

−10.48756514291918225880809977724, −9.187988855067739667654742356537, −9.033362056149468604725474770848, −8.110230441796504161448911277044, −7.34728518749395822536217945552, −5.93546433603801149343309415320, −5.39600011450470021064902058397, −3.48423005391246629187370382559, −2.11281706394383952564981573378, −1.32684326948215444633697507164, 0.55486566749689323584686418394, 2.92984752090671733113679435277, 4.12769314759448234400913810229, 4.85806875993636940540660842388, 6.44915423314798490153715964359, 7.28014406354453607960488881880, 7.84310887395790052788090712614, 8.999405387271892159282642500864, 9.640797715676311361428452163596, 10.22460602812569607061345539470

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