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Properties

Label 2-1550-155.129-c1-0-46
Degree $2$
Conductor $1550$
Sign $-0.759 - 0.650i$
Analytic cond. $12.3768$
Root an. cond. $3.51806$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (0.866 + 0.5i)3-s − 4-s + (0.5 − 0.866i)6-s + (0.866 + 0.5i)7-s + i·8-s + (−1 − 1.73i)9-s + (−1.5 − 2.59i)11-s + (−0.866 − 0.5i)12-s + (−4.33 + 2.5i)13-s + (0.5 − 0.866i)14-s + 16-s + (−2.59 − 1.5i)17-s + (−1.73 + i)18-s + (−3.5 + 6.06i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.499 + 0.288i)3-s − 0.5·4-s + (0.204 − 0.353i)6-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.333 − 0.577i)9-s + (−0.452 − 0.783i)11-s + (−0.249 − 0.144i)12-s + (−1.20 + 0.693i)13-s + (0.133 − 0.231i)14-s + 0.250·16-s + (−0.630 − 0.363i)17-s + (−0.408 + 0.235i)18-s + (−0.802 + 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1550\)    =    \(2 \cdot 5^{2} \cdot 31\)
Sign: $-0.759 - 0.650i$
Analytic conductor: \(12.3768\)
Root analytic conductor: \(3.51806\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1550} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1550,\ (\ :1/2),\ -0.759 - 0.650i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
31 \( 1 + (2 - 5.19i)T \)
good3 \( 1 + (-0.866 - 0.5i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.33 - 2.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.59 + 1.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
37 \( 1 + (-6.06 - 3.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.33 - 2.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + (7.79 - 4.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + (11.2 - 6.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.5 - 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-11.2 + 6.5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.79 + 4.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.050133019398102179502571843807, −8.411507278648937130416585141480, −7.60058121389509469049801881583, −6.43937015299395863658424686031, −5.50548531292308467014568785590, −4.54050453330775112721863154012, −3.67913420818630152261207894454, −2.76289046709465694294026776076, −1.83019899775164290906252151275, 0, 2.03916536742807625926313588118, 2.83474652433311565140558379908, 4.39474656752294989892279453545, 4.90670875635124646282194689433, 5.89051520460781965306758479134, 6.93816700545911497925862044436, 7.77541004423454332330771047849, 7.899090518560633157924967188072, 9.131653289916896860633340548933

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