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Properties

Label 2-1254-19.11-c1-0-11
Degree $2$
Conductor $1254$
Sign $-0.476 - 0.879i$
Analytic cond. $10.0132$
Root an. cond. $3.16437$
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 + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.436 + 0.756i)5-s + (−0.499 − 0.866i)6-s + 2.30·7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.436 − 0.756i)10-s + 11-s + 0.999·12-s + (−1.83 − 3.17i)13-s + (−1.15 + 1.99i)14-s + (−0.436 − 0.756i)15-s + (−0.5 + 0.866i)16-s + (−2.77 + 4.79i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.195 + 0.338i)5-s + (−0.204 − 0.353i)6-s + 0.870·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.138 − 0.239i)10-s + 0.301·11-s + 0.288·12-s + (−0.508 − 0.881i)13-s + (−0.307 + 0.533i)14-s + (−0.112 − 0.195i)15-s + (−0.125 + 0.216i)16-s + (−0.672 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1254\)    =    \(2 \cdot 3 \cdot 11 \cdot 19\)
Sign: $-0.476 - 0.879i$
Analytic conductor: \(10.0132\)
Root analytic conductor: \(3.16437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1254} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1254,\ (\ :1/2),\ -0.476 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.166871814\)
\(L(\frac12)\) \(\approx\) \(1.166871814\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 - T \)
19 \( 1 + (-4.23 - 1.03i)T \)
good5 \( 1 + (0.436 - 0.756i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 2.30T + 7T^{2} \)
13 \( 1 + (1.83 + 3.17i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.77 - 4.79i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.89 + 3.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.41 - 7.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.71T + 31T^{2} \)
37 \( 1 + 1.49T + 37T^{2} \)
41 \( 1 + (-2.77 + 4.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0634 + 0.109i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.22 - 3.86i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.29 - 2.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.15 - 7.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.391 - 0.678i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.16 + 3.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.79 - 10.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.30 - 2.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.77 - 13.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (-4.69 - 8.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.84 + 6.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.12063570514000795019106736738, −8.968221857763368764727459873110, −8.359325808176145017148844365715, −7.55691901498429020810210695536, −6.70655034655967476274109250096, −5.77428367192249785784222976488, −4.96859012034291835806392943369, −4.15551674622343546962835983254, −2.88208773520331954159130395024, −1.23333601510744407328375194367, 0.67395903588777072062469894108, 1.86493820082782345531524465689, 2.88593477479503770510195898043, 4.48958876521336589694027345140, 4.80256821665096209033630333578, 6.18971431954536265152756735952, 7.15206159043781804287333645758, 7.87003326948656080697466217077, 8.632685003938375102203008159525, 9.467318284915433496960246118871

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