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Properties

Label 2-950-19.17-c1-0-17
Degree $2$
Conductor $950$
Sign $0.926 + 0.376i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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  + (0.173 − 0.984i)2-s + (1.28 + 1.07i)3-s + (−0.939 − 0.342i)4-s + (1.28 − 1.07i)6-s + (1.16 + 2.01i)7-s + (−0.5 + 0.866i)8-s + (−0.0339 − 0.192i)9-s + (1.54 − 2.68i)11-s + (−0.837 − 1.45i)12-s + (5.18 − 4.35i)13-s + (2.18 − 0.794i)14-s + (0.766 + 0.642i)16-s + (−0.818 + 4.64i)17-s − 0.195·18-s + (−0.846 + 4.27i)19-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (0.740 + 0.621i)3-s + (−0.469 − 0.171i)4-s + (0.523 − 0.439i)6-s + (0.439 + 0.760i)7-s + (−0.176 + 0.306i)8-s + (−0.0113 − 0.0641i)9-s + (0.466 − 0.808i)11-s + (−0.241 − 0.418i)12-s + (1.43 − 1.20i)13-s + (0.583 − 0.212i)14-s + (0.191 + 0.160i)16-s + (−0.198 + 1.12i)17-s − 0.0460·18-s + (−0.194 + 0.980i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.926 + 0.376i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.926 + 0.376i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25388 - 0.440174i\)
\(L(\frac12)\) \(\approx\) \(2.25388 - 0.440174i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 \)
19 \( 1 + (0.846 - 4.27i)T \)
good3 \( 1 + (-1.28 - 1.07i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (-1.16 - 2.01i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.54 + 2.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.18 + 4.35i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.818 - 4.64i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (2.99 + 1.09i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.115 - 0.653i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.25 - 3.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.57T + 37T^{2} \)
41 \( 1 + (-1.86 - 1.56i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.05 + 0.748i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.08 + 6.18i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-9.43 - 3.43i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.77 - 10.0i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (3.49 + 1.27i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.920 + 5.22i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (11.0 - 4.03i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (1.96 + 1.64i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (12.6 + 10.6i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (3.21 + 5.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.47 - 5.43i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (0.923 - 5.23i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.18976902111356776314881701774, −8.988577648044967599129514310031, −8.550341653209688267139432239706, −8.046706243996791733931898706101, −6.10019718830196744823428791313, −5.76951583108593686220866387576, −4.23060334211888953228157770661, −3.60231187324275467462648393234, −2.72017301671392999551518497938, −1.28871083592829553177946925621, 1.29660949284174114005944764516, 2.56334330545189540249514742466, 4.08249706010751393357981180972, 4.60019003657776759035670308070, 6.03375590443735344431421800520, 6.97060610728549946473007177453, 7.41506811561861406166717823513, 8.300434376818548572693314550479, 9.045374661896974319877434039809, 9.754801453212961125213999100955

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