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Properties

Label 2-250-125.106-c1-0-1
Degree $2$
Conductor $250$
Sign $-0.203 - 0.979i$
Analytic cond. $1.99626$
Root an. cond. $1.41289$
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  + (0.535 − 0.844i)2-s + (−0.665 + 0.625i)3-s + (−0.425 − 0.904i)4-s + (−1.31 + 1.81i)5-s + (0.171 + 0.897i)6-s + (−3.97 + 2.88i)7-s + (−0.992 − 0.125i)8-s + (−0.135 + 2.16i)9-s + (0.825 + 2.07i)10-s + (−0.806 + 1.27i)11-s + (0.849 + 0.336i)12-s + (0.369 − 5.87i)13-s + (0.308 + 4.90i)14-s + (−0.258 − 2.02i)15-s + (−0.637 + 0.770i)16-s + (−0.249 + 0.530i)17-s + ⋯
L(s)  = 1  + (0.378 − 0.597i)2-s + (−0.384 + 0.360i)3-s + (−0.212 − 0.452i)4-s + (−0.586 + 0.809i)5-s + (0.0698 + 0.366i)6-s + (−1.50 + 1.09i)7-s + (−0.350 − 0.0443i)8-s + (−0.0453 + 0.720i)9-s + (0.261 + 0.657i)10-s + (−0.243 + 0.383i)11-s + (0.245 + 0.0970i)12-s + (0.102 − 1.62i)13-s + (0.0824 + 1.31i)14-s + (−0.0666 − 0.523i)15-s + (−0.159 + 0.192i)16-s + (−0.0605 + 0.128i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $-0.203 - 0.979i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ -0.203 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.430537 + 0.528955i\)
\(L(\frac12)\) \(\approx\) \(0.430537 + 0.528955i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.535 + 0.844i)T \)
5 \( 1 + (1.31 - 1.81i)T \)
good3 \( 1 + (0.665 - 0.625i)T + (0.188 - 2.99i)T^{2} \)
7 \( 1 + (3.97 - 2.88i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.806 - 1.27i)T + (-4.68 - 9.95i)T^{2} \)
13 \( 1 + (-0.369 + 5.87i)T + (-12.8 - 1.62i)T^{2} \)
17 \( 1 + (0.249 - 0.530i)T + (-10.8 - 13.0i)T^{2} \)
19 \( 1 + (-3.55 - 3.34i)T + (1.19 + 18.9i)T^{2} \)
23 \( 1 + (-3.00 - 0.770i)T + (20.1 + 11.0i)T^{2} \)
29 \( 1 + (-0.811 - 0.446i)T + (15.5 + 24.4i)T^{2} \)
31 \( 1 + (3.68 - 7.83i)T + (-19.7 - 23.8i)T^{2} \)
37 \( 1 + (6.07 - 7.34i)T + (-6.93 - 36.3i)T^{2} \)
41 \( 1 + (-7.08 + 1.81i)T + (35.9 - 19.7i)T^{2} \)
43 \( 1 + (0.307 - 0.946i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-0.824 + 0.104i)T + (45.5 - 11.6i)T^{2} \)
53 \( 1 + (0.505 - 2.64i)T + (-49.2 - 19.5i)T^{2} \)
59 \( 1 + (-0.315 - 0.124i)T + (43.0 + 40.3i)T^{2} \)
61 \( 1 + (-7.74 - 1.98i)T + (53.4 + 29.3i)T^{2} \)
67 \( 1 + (0.770 - 0.423i)T + (35.9 - 56.5i)T^{2} \)
71 \( 1 + (3.88 - 0.491i)T + (68.7 - 17.6i)T^{2} \)
73 \( 1 + (3.90 - 1.54i)T + (53.2 - 49.9i)T^{2} \)
79 \( 1 + (7.71 - 7.24i)T + (4.96 - 78.8i)T^{2} \)
83 \( 1 + (-5.58 - 5.24i)T + (5.21 + 82.8i)T^{2} \)
89 \( 1 + (5.28 - 2.09i)T + (64.8 - 60.9i)T^{2} \)
97 \( 1 + (10.1 + 5.59i)T + (51.9 + 81.8i)T^{2} \)
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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

−12.39125845655088376212324170408, −11.37457346295465123759841824578, −10.35375977152998452523039467620, −9.980169290170976704554424992948, −8.567603077592624905184176669555, −7.27518905368593053899977157417, −5.96663531292668119544973594733, −5.19258769109467313085191298583, −3.45939239769248468818052922187, −2.75193825566282486392140557056, 0.49149589656980842999794423685, 3.48759957406169021910366147459, 4.34709095180661966328797464446, 5.82952161742299939890422577787, 6.86836489672480485921679450246, 7.37719096438867792233389579214, 9.010023646903320205552947584660, 9.498060547260388540467540715403, 11.14353139475570288902291698279, 11.97048216594375648695399559791

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