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Properties

Label 2-661-661.247-c1-0-52
Degree $2$
Conductor $661$
Sign $-0.179 + 0.983i$
Analytic cond. $5.27811$
Root an. cond. $2.29741$
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  + (0.778 + 0.565i)2-s + 1.01·3-s + (−0.331 − 1.02i)4-s + (0.631 − 1.94i)5-s + (0.791 + 0.575i)6-s − 3.88·7-s + (0.914 − 2.81i)8-s − 1.96·9-s + (1.59 − 1.15i)10-s − 3.01·11-s + (−0.337 − 1.03i)12-s + (−0.195 + 0.600i)13-s + (−3.02 − 2.19i)14-s + (0.641 − 1.97i)15-s + (0.565 − 0.410i)16-s + (1.22 − 3.76i)17-s + ⋯
L(s)  = 1  + (0.550 + 0.399i)2-s + 0.586·3-s + (−0.165 − 0.510i)4-s + (0.282 − 0.868i)5-s + (0.323 + 0.234i)6-s − 1.46·7-s + (0.323 − 0.994i)8-s − 0.655·9-s + (0.502 − 0.365i)10-s − 0.907·11-s + (−0.0973 − 0.299i)12-s + (−0.0541 + 0.166i)13-s + (−0.807 − 0.586i)14-s + (0.165 − 0.509i)15-s + (0.141 − 0.102i)16-s + (0.296 − 0.913i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 661 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 661 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(661\)
Sign: $-0.179 + 0.983i$
Analytic conductor: \(5.27811\)
Root analytic conductor: \(2.29741\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{661} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 661,\ (\ :1/2),\ -0.179 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.995621 - 1.19330i\)
\(L(\frac12)\) \(\approx\) \(0.995621 - 1.19330i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad661 \( 1 + (-18.3 + 17.9i)T \)
good2 \( 1 + (-0.778 - 0.565i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 - 1.01T + 3T^{2} \)
5 \( 1 + (-0.631 + 1.94i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + 3.88T + 7T^{2} \)
11 \( 1 + 3.01T + 11T^{2} \)
13 \( 1 + (0.195 - 0.600i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.22 + 3.76i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.63 + 4.09i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.76 - 1.28i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 - 0.302T + 29T^{2} \)
31 \( 1 + (0.304 - 0.220i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.33 + 4.10i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.35 - 2.43i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-0.744 - 0.541i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-9.84 + 7.15i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.03 - 0.750i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.938 + 2.88i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.28 + 6.02i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-7.79 - 5.66i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-5.57 - 4.04i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.10 + 12.6i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.99 - 6.13i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.91 - 15.1i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 1.62T + 89T^{2} \)
97 \( 1 + (-3.50 - 2.54i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.890591721932399109854673483322, −9.435807424345031001709561782279, −8.806151677793626264016903847119, −7.48015181516876311868877131359, −6.65581086380348916756795468935, −5.42760678780296843630218175071, −5.16589554881858187289065173094, −3.62536497555512034931392307922, −2.66749921349396002471474439366, −0.62478381198814169707840870617, 2.49390442870009489232652018697, 3.11469708473436773055565811666, 3.71741629464728337083911977799, 5.34954180545165817100239171804, 6.19071329147279356462031614499, 7.35018220665877000565316933868, 8.129440188100223503217645562717, 9.066297586616452098258364958310, 10.07346895164008374539717966199, 10.65088897232701370382677357874

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