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Properties

Label 2-250-25.14-c1-0-6
Degree $2$
Conductor $250$
Sign $0.545 - 0.837i$
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  + (0.587 + 0.809i)2-s + (2.48 + 0.809i)3-s + (−0.309 + 0.951i)4-s + (0.809 + 2.48i)6-s − 3i·7-s + (−0.951 + 0.309i)8-s + (3.11 + 2.26i)9-s + (0.190 − 0.138i)11-s + (−1.53 + 2.11i)12-s + (−0.587 + 0.809i)13-s + (2.42 − 1.76i)14-s + (−0.809 − 0.587i)16-s + (−7.46 + 2.42i)17-s + 3.85i·18-s + (0.263 + 0.812i)19-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (1.43 + 0.467i)3-s + (−0.154 + 0.475i)4-s + (0.330 + 1.01i)6-s − 1.13i·7-s + (−0.336 + 0.109i)8-s + (1.03 + 0.755i)9-s + (0.0575 − 0.0418i)11-s + (−0.444 + 0.611i)12-s + (−0.163 + 0.224i)13-s + (0.648 − 0.471i)14-s + (−0.202 − 0.146i)16-s + (−1.81 + 0.588i)17-s + 0.908i·18-s + (0.0605 + 0.186i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $0.545 - 0.837i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ 0.545 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91471 + 1.03782i\)
\(L(\frac12)\) \(\approx\) \(1.91471 + 1.03782i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 \)
good3 \( 1 + (-2.48 - 0.809i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + (-0.190 + 0.138i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.587 - 0.809i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (7.46 - 2.42i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.263 - 0.812i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.66 + 5.04i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.163 + 0.502i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.30 - 4.02i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.30 + 5.92i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.04 - 4.39i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.76iT - 43T^{2} \)
47 \( 1 + (-5.65 - 1.83i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.45 + 0.472i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.61 - 2.62i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.73 + 1.26i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-5.48 + 1.78i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.927 - 2.85i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.35 + 4.61i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.854 + 2.62i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (12.8 - 4.16i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (3.61 - 2.62i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (10.0 + 3.26i)T + (78.4 + 57.0i)T^{2} \)
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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

−12.65089619838764144027672376059, −11.09515658539362515344516312723, −10.13480054423073644182023642279, −9.072957605641889532860809694062, −8.303533088658632641327973487956, −7.38068811781648472916061623172, −6.34579048851586895464574287368, −4.43938634878455591738178291575, −3.95034299686036942159072292257, −2.46915054419044815812603744465, 2.09378425469208818846876058417, 2.81591016590616022669371027351, 4.21614085590050743232153156815, 5.68216944181054893102906862252, 7.02705616815833124479476114916, 8.236013497482302924268703663574, 9.047935214567123063741663089381, 9.696507287063433945384570717883, 11.19650298510664816496616030834, 12.03911406645517335584845533798

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