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Properties

Label 2-650-13.10-c1-0-0
Degree $2$
Conductor $650$
Sign $-0.265 - 0.964i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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  + (−0.866 + 0.5i)2-s + (−1.36 − 2.36i)3-s + (0.499 − 0.866i)4-s + (2.36 + 1.36i)6-s + (2.59 + 1.5i)7-s + 0.999i·8-s + (−2.23 + 3.86i)9-s + (−2.59 + 1.5i)11-s − 2.73·12-s + (−3.5 − 0.866i)13-s − 3·14-s + (−0.5 − 0.866i)16-s + (−1.09 + 1.90i)17-s − 4.46i·18-s + (−5.59 − 3.23i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.788 − 1.36i)3-s + (0.249 − 0.433i)4-s + (0.965 + 0.557i)6-s + (0.981 + 0.566i)7-s + 0.353i·8-s + (−0.744 + 1.28i)9-s + (−0.783 + 0.452i)11-s − 0.788·12-s + (−0.970 − 0.240i)13-s − 0.801·14-s + (−0.125 − 0.216i)16-s + (−0.266 + 0.461i)17-s − 1.05i·18-s + (−1.28 − 0.741i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.265 - 0.964i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.169133 + 0.221888i\)
\(L(\frac12)\) \(\approx\) \(0.169133 + 0.221888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (3.5 + 0.866i)T \)
good3 \( 1 + (1.36 + 2.36i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.09 - 1.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.59 + 3.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.26 - 2.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.73 - 8.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 + (9.69 - 5.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (9 - 5.19i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 - 6.46T + 53T^{2} \)
59 \( 1 + (-9 - 5.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.09 + 3.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.66iT - 73T^{2} \)
79 \( 1 - 6.19T + 79T^{2} \)
83 \( 1 + 2.19iT - 83T^{2} \)
89 \( 1 + (14.8 - 8.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.0 + 7.56i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.84678319203909862077085547394, −10.10610322128751143062267084694, −8.595547886037132185282116899596, −8.225468643439661416296446538977, −7.06314965287323940740848459550, −6.76765385659634170862555125782, −5.39401755039797036154114010828, −4.95893930195679783831865547853, −2.45983518447084707004709035655, −1.54570428681967192446307665067, 0.19707014519970794248426136296, 2.30020745170218817334942387266, 3.88315693059237228642654652717, 4.66351034631059840536664180102, 5.47717172872932225656394369965, 6.76824294692777519608279926679, 7.925438361058003848818379656596, 8.687451506994531202531981515261, 9.768345780695713729655913253303, 10.44208464592973855366798774936

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