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Properties

Label 2-66-11.8-c2-0-1
Degree $2$
Conductor $66$
Sign $0.218 - 0.975i$
Analytic cond. $1.79836$
Root an. cond. $1.34103$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.831 + 1.14i)2-s + (0.535 + 1.64i)3-s + (−0.618 + 1.90i)4-s + (1.74 + 1.26i)5-s + (−1.43 + 1.98i)6-s + (−1.27 − 0.414i)7-s + (−2.68 + 0.874i)8-s + (−2.42 + 1.76i)9-s + 3.05i·10-s + (10.9 − 0.273i)11-s − 3.46·12-s + (−6.36 − 8.75i)13-s + (−0.586 − 1.80i)14-s + (−1.15 + 3.55i)15-s + (−3.23 − 2.35i)16-s + (15.8 − 21.8i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (0.178 + 0.549i)3-s + (−0.154 + 0.475i)4-s + (0.349 + 0.253i)5-s + (−0.239 + 0.330i)6-s + (−0.182 − 0.0592i)7-s + (−0.336 + 0.109i)8-s + (−0.269 + 0.195i)9-s + 0.305i·10-s + (0.999 − 0.0248i)11-s − 0.288·12-s + (−0.489 − 0.673i)13-s + (−0.0418 − 0.128i)14-s + (−0.0770 + 0.237i)15-s + (−0.202 − 0.146i)16-s + (0.934 − 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.218 - 0.975i$
Analytic conductor: \(1.79836\)
Root analytic conductor: \(1.34103\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1),\ 0.218 - 0.975i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.21728 + 0.975370i\)
\(L(\frac12)\) \(\approx\) \(1.21728 + 0.975370i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 - 1.14i)T \)
3 \( 1 + (-0.535 - 1.64i)T \)
11 \( 1 + (-10.9 + 0.273i)T \)
good5 \( 1 + (-1.74 - 1.26i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (1.27 + 0.414i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (6.36 + 8.75i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (-15.8 + 21.8i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (9.95 - 3.23i)T + (292. - 212. i)T^{2} \)
23 \( 1 - 14.1T + 529T^{2} \)
29 \( 1 + (11.7 + 3.81i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (20.7 - 15.0i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (9.00 - 27.7i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (10.1 - 3.30i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 41.5iT - 1.84e3T^{2} \)
47 \( 1 + (-21.6 - 66.4i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (48.5 - 35.2i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (16.2 - 49.8i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-64.7 + 89.1i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 60.0T + 4.48e3T^{2} \)
71 \( 1 + (56.0 + 40.7i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (69.7 + 22.6i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (58.9 + 81.1i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-59.8 + 82.4i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 55.9T + 7.92e3T^{2} \)
97 \( 1 + (-141. + 102. i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−14.65794332210257025340234271253, −14.14209504031607768811702426133, −12.81121846537311203693517228683, −11.59947499896954422477494001094, −10.11932791815729893663673410771, −9.108381496830913824485222728701, −7.61296677738214733667223521558, −6.23995534066607759909247510178, −4.83699347174514588144832995978, −3.18652320949389420922869901358, 1.76591456595183326505035563281, 3.79824742498548938170444173454, 5.64107509383193057255279498015, 6.98223332778320233681194103427, 8.738399944649011583485584911294, 9.782828560448600805406801737550, 11.23656682056512357054467529977, 12.33930146403328103187368958235, 13.09049425715115621439134779059, 14.26950169335982953785321566404

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