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Properties

Label 2-250-25.19-c1-0-2
Degree $2$
Conductor $250$
Sign $0.982 - 0.184i$
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  + (0.951 − 0.309i)2-s + (0.792 + 1.09i)3-s + (0.809 − 0.587i)4-s + (1.09 + 0.792i)6-s + 0.833i·7-s + (0.587 − 0.809i)8-s + (0.365 − 1.12i)9-s + (0.257 + 0.792i)11-s + (1.28 + 0.416i)12-s + (4.34 + 1.41i)13-s + (0.257 + 0.792i)14-s + (0.309 − 0.951i)16-s + (−3.20 + 4.41i)17-s − 1.18i·18-s + (−7.00 − 5.08i)19-s + ⋯
L(s)  = 1  + (0.672 − 0.218i)2-s + (0.457 + 0.629i)3-s + (0.404 − 0.293i)4-s + (0.445 + 0.323i)6-s + 0.314i·7-s + (0.207 − 0.286i)8-s + (0.121 − 0.374i)9-s + (0.0776 + 0.238i)11-s + (0.370 + 0.120i)12-s + (1.20 + 0.391i)13-s + (0.0688 + 0.211i)14-s + (0.0772 − 0.237i)16-s + (−0.777 + 1.06i)17-s − 0.278i·18-s + (−1.60 − 1.16i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $0.982 - 0.184i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ 0.982 - 0.184i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05091 + 0.191022i\)
\(L(\frac12)\) \(\approx\) \(2.05091 + 0.191022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 \)
good3 \( 1 + (-0.792 - 1.09i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 - 0.833iT - 7T^{2} \)
11 \( 1 + (-0.257 - 0.792i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-4.34 - 1.41i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.20 - 4.41i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (7.00 + 5.08i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.35 - 1.09i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.64 + 1.92i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.85 + 3.52i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (6.95 + 2.26i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.576 + 1.77i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.63iT - 43T^{2} \)
47 \( 1 + (0.489 + 0.674i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.77 + 5.19i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.18 - 12.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.81 - 5.59i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (0.881 - 1.21i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-1.91 + 1.38i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.16 + 1.02i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.18 + 3.03i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.25 - 9.97i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.16 - 6.66i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-6.51 - 8.97i)T + (-29.9 + 92.2i)T^{2} \)
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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

−12.18420968836303179300073227164, −11.06865802827866104328346715429, −10.38171981962777799581765625094, −9.104921264518430985337356762651, −8.520538798956664151276721275398, −6.79633966175979400280367104818, −5.96445336578867042438924322829, −4.38700439935009846328193400867, −3.75146624890653371441375299953, −2.15187283803703278951908921365, 1.90350793421377299155718746747, 3.41488920989453613889296688081, 4.66905877408563103191611728895, 6.07250709250268051423149517371, 6.95324472128592976649235627512, 8.054593135154012468745841168122, 8.735722289875921410346235373636, 10.40671922033038354439232530851, 11.10461745952502447329977921997, 12.39586475437539298263668615728

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