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Properties

Label 2-650-65.9-c1-0-19
Degree $2$
Conductor $650$
Sign $-0.708 + 0.705i$
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 + (0.633 + 0.366i)3-s + (0.499 + 0.866i)4-s + (−0.366 − 0.633i)6-s + (−2.13 + 1.23i)7-s − 0.999i·8-s + (−1.23 − 2.13i)9-s + (0.866 − 1.5i)11-s + 0.732i·12-s + (−3.23 − 1.59i)13-s + 2.46·14-s + (−0.5 + 0.866i)16-s + (−1.09 + 0.633i)17-s + 2.46i·18-s + (−1.59 − 2.76i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.366 + 0.211i)3-s + (0.249 + 0.433i)4-s + (−0.149 − 0.258i)6-s + (−0.806 + 0.465i)7-s − 0.353i·8-s + (−0.410 − 0.711i)9-s + (0.261 − 0.452i)11-s + 0.211i·12-s + (−0.896 − 0.443i)13-s + 0.658·14-s + (−0.125 + 0.216i)16-s + (−0.266 + 0.153i)17-s + 0.580i·18-s + (−0.366 − 0.635i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.708 + 0.705i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.708 + 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.210432 - 0.509440i\)
\(L(\frac12)\) \(\approx\) \(0.210432 - 0.509440i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.23 + 1.59i)T \)
good3 \( 1 + (-0.633 - 0.366i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.13 - 1.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.866 + 1.5i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.09 - 0.633i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.59 + 2.76i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 1.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.73 + 8.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.26T + 31T^{2} \)
37 \( 1 + (7.96 + 4.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.46 - 6i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.73 + 4.46i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.46iT - 47T^{2} \)
53 \( 1 + 1.73iT - 53T^{2} \)
59 \( 1 + (-1.26 - 2.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.36 + 5.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.26 + 4.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.73 + 13.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.73iT - 73T^{2} \)
79 \( 1 + 3.26T + 79T^{2} \)
83 \( 1 - 11.6iT - 83T^{2} \)
89 \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.56 + 2.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.02265366763474680519563600303, −9.375745866724845900464239845509, −8.705126718395217326171229289424, −7.86753029070321972287621800985, −6.62644326685147171788303766793, −5.95417671111637243969054440334, −4.40280314703840360383884558710, −3.20845062938873076849704284101, −2.44038524152103582103005042475, −0.32607077181739384323053018558, 1.78429498265555275235997333341, 3.02400690288632082222228042366, 4.45242077332325339672147592411, 5.59364345770718273381349869033, 6.82088500255322820256992485820, 7.25349026123573436107713123469, 8.330156712519497574287376732877, 9.030487933034131648923965346071, 10.07081530253443678273843795078, 10.45104667658008519914461835936

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