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Properties

Label 2-890-5.4-c1-0-36
Degree $2$
Conductor $890$
Sign $-0.981 + 0.190i$
Analytic cond. $7.10668$
Root an. cond. $2.66583$
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  − i·2-s + 1.57i·3-s − 4-s + (−0.425 − 2.19i)5-s + 1.57·6-s − 0.668i·7-s + i·8-s + 0.531·9-s + (−2.19 + 0.425i)10-s − 4.49·11-s − 1.57i·12-s − 3.72i·13-s − 0.668·14-s + (3.44 − 0.668i)15-s + 16-s + 4.57i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.907i·3-s − 0.5·4-s + (−0.190 − 0.981i)5-s + 0.641·6-s − 0.252i·7-s + 0.353i·8-s + 0.177·9-s + (−0.694 + 0.134i)10-s − 1.35·11-s − 0.453i·12-s − 1.03i·13-s − 0.178·14-s + (0.890 − 0.172i)15-s + 0.250·16-s + 1.10i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(890\)    =    \(2 \cdot 5 \cdot 89\)
Sign: $-0.981 + 0.190i$
Analytic conductor: \(7.10668\)
Root analytic conductor: \(2.66583\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{890} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 890,\ (\ :1/2),\ -0.981 + 0.190i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0542665 - 0.564866i\)
\(L(\frac12)\) \(\approx\) \(0.0542665 - 0.564866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (0.425 + 2.19i)T \)
89 \( 1 - T \)
good3 \( 1 - 1.57iT - 3T^{2} \)
7 \( 1 + 0.668iT - 7T^{2} \)
11 \( 1 + 4.49T + 11T^{2} \)
13 \( 1 + 3.72iT - 13T^{2} \)
17 \( 1 - 4.57iT - 17T^{2} \)
19 \( 1 + 4.51T + 19T^{2} \)
23 \( 1 + 6.82iT - 23T^{2} \)
29 \( 1 - 2.09T + 29T^{2} \)
31 \( 1 + 7.75T + 31T^{2} \)
37 \( 1 + 8.18iT - 37T^{2} \)
41 \( 1 + 9.01T + 41T^{2} \)
43 \( 1 + 2.06iT - 43T^{2} \)
47 \( 1 + 0.855iT - 47T^{2} \)
53 \( 1 + 2.10iT - 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 3.74T + 61T^{2} \)
67 \( 1 + 14.0iT - 67T^{2} \)
71 \( 1 + 5.86T + 71T^{2} \)
73 \( 1 - 9.44iT - 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + 8.29iT - 83T^{2} \)
97 \( 1 + 3.14iT - 97T^{2} \)
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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

−10.07994111871179586369002532546, −8.887361937442107135533266687353, −8.396431767993877244464804417895, −7.46874855734013373355857421186, −5.84269932682086114546990657267, −5.00011349919110372500821040352, −4.30212319740492098189663575287, −3.42732225764572992416397258868, −2.00086535223430625349671817082, −0.25797487011315823431978811572, 1.90618071371158179176056092994, 3.04598717906501548602664451420, 4.41075171475191218416195597726, 5.52000441703150355074213843918, 6.49050037715854957160868724862, 7.18295692403405537991588031052, 7.62524662370931654171617655460, 8.575699769504953752449033097312, 9.646755307740579073970289482404, 10.43262133363897121997189352898

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