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Properties

Label 2-266-19.7-c1-0-2
Degree $2$
Conductor $266$
Sign $0.963 - 0.267i$
Analytic cond. $2.12402$
Root an. cond. $1.45740$
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  + (0.5 + 0.866i)2-s + (−1.51 − 2.62i)3-s + (−0.499 + 0.866i)4-s + (1.68 + 2.91i)5-s + (1.51 − 2.62i)6-s + 7-s − 0.999·8-s + (−3.08 + 5.34i)9-s + (−1.68 + 2.91i)10-s + 5.02·11-s + 3.02·12-s + (2.41 − 4.18i)13-s + (0.5 + 0.866i)14-s + (5.10 − 8.83i)15-s + (−0.5 − 0.866i)16-s + (0.0733 + 0.127i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.874 − 1.51i)3-s + (−0.249 + 0.433i)4-s + (0.753 + 1.30i)5-s + (0.618 − 1.07i)6-s + 0.377·7-s − 0.353·8-s + (−1.02 + 1.78i)9-s + (−0.532 + 0.922i)10-s + 1.51·11-s + 0.874·12-s + (0.670 − 1.16i)13-s + (0.133 + 0.231i)14-s + (1.31 − 2.28i)15-s + (−0.125 − 0.216i)16-s + (0.0177 + 0.0308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 266 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(266\)    =    \(2 \cdot 7 \cdot 19\)
Sign: $0.963 - 0.267i$
Analytic conductor: \(2.12402\)
Root analytic conductor: \(1.45740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{266} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 266,\ (\ :1/2),\ 0.963 - 0.267i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35455 + 0.184415i\)
\(L(\frac12)\) \(\approx\) \(1.35455 + 0.184415i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 - T \)
19 \( 1 + (-3.85 + 2.02i)T \)
good3 \( 1 + (1.51 + 2.62i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.68 - 2.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 5.02T + 11T^{2} \)
13 \( 1 + (-2.41 + 4.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.0733 - 0.127i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.28 - 3.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.76 - 6.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 + 4.73T + 37T^{2} \)
41 \( 1 + (-0.485 - 0.840i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.26 + 3.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.76 + 6.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.26 + 2.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.27 + 2.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.52 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.441 - 0.764i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.645 + 1.11i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.92 + 8.53i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.20 + 14.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 + (0.970 - 1.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.54 + 9.60i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.97382433572201452773648628336, −11.34248149396320432933906087947, −10.37384792431062391172599400850, −8.849975294045505003734103163793, −7.45806774623645213492255191113, −6.98785036082064040843736660325, −6.05933959299882112815635367019, −5.48476965951310828513511241012, −3.30946286701857386784484683666, −1.59072246803545633330961511335, 1.41563146425643788841303719244, 3.94440490362706695995611126400, 4.46881037940284154613903499474, 5.52535477200413539906257587828, 6.28405811542751535473376449798, 8.714488556366982864637313735299, 9.370034939792255746397577844229, 9.900593415158260541257975954159, 11.15096124329519209755538526000, 11.72973459828120692279465200685

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