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Properties

Label 2-650-25.11-c1-0-2
Degree $2$
Conductor $650$
Sign $-0.0206 - 0.999i$
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  + (0.809 − 0.587i)2-s + (−0.311 + 0.960i)3-s + (0.309 − 0.951i)4-s + (−2.22 + 0.233i)5-s + (0.311 + 0.960i)6-s − 0.400·7-s + (−0.309 − 0.951i)8-s + (1.60 + 1.16i)9-s + (−1.66 + 1.49i)10-s + (−3.50 + 2.54i)11-s + (0.816 + 0.593i)12-s + (0.809 + 0.587i)13-s + (−0.324 + 0.235i)14-s + (0.469 − 2.20i)15-s + (−0.809 − 0.587i)16-s + (1.02 + 3.14i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.180 + 0.554i)3-s + (0.154 − 0.475i)4-s + (−0.994 + 0.104i)5-s + (0.127 + 0.391i)6-s − 0.151·7-s + (−0.109 − 0.336i)8-s + (0.534 + 0.388i)9-s + (−0.525 + 0.473i)10-s + (−1.05 + 0.766i)11-s + (0.235 + 0.171i)12-s + (0.224 + 0.163i)13-s + (−0.0866 + 0.0629i)14-s + (0.121 − 0.570i)15-s + (−0.202 − 0.146i)16-s + (0.248 + 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0206 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0206 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.0206 - 0.999i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.0206 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.816107 + 0.833165i\)
\(L(\frac12)\) \(\approx\) \(0.816107 + 0.833165i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (2.22 - 0.233i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (0.311 - 0.960i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 0.400T + 7T^{2} \)
11 \( 1 + (3.50 - 2.54i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (-1.02 - 3.14i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.63 - 5.01i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.42 - 3.21i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.40 - 7.38i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.54 + 7.82i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.58 + 2.60i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.60 - 5.52i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 + (-0.115 + 0.356i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.968 + 2.98i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.12 - 1.54i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (6.95 - 5.05i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.29 + 10.1i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.974 + 3.00i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-11.6 + 8.48i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.55 - 7.86i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.171 + 0.527i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-14.5 + 10.5i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.961 - 2.95i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.75013992573974074318447328762, −10.20307825566116416208055556871, −9.331247734140510691071501473271, −7.84212105097278781213080738023, −7.48805544241332256636673071782, −6.02328791666546400016374970726, −5.06394625402122012618952716393, −4.14182482325432704870577391854, −3.45477044361098070239073559906, −1.87684819629777063975174580453, 0.52945545925845163484689775164, 2.70714883165405939312202811560, 3.77804168583553688533509583824, 4.82712538556869316303694159768, 5.83762751958281168714862912496, 6.86683587144083810990552586576, 7.55737698938383873109375563342, 8.249706188703391956507506914422, 9.300412942757988391715295910710, 10.58601441165867234054146032477

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