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Properties

Label 2-950-19.9-c1-0-27
Degree $2$
Conductor $950$
Sign $0.518 + 0.855i$
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 + (2.12 − 1.78i)3-s + (−0.939 + 0.342i)4-s + (2.12 + 1.78i)6-s + (0.303 − 0.526i)7-s + (−0.5 − 0.866i)8-s + (0.817 − 4.63i)9-s + (−1.15 − 2.00i)11-s + (−1.38 + 2.40i)12-s + (−5.07 − 4.25i)13-s + (0.570 + 0.207i)14-s + (0.766 − 0.642i)16-s + (−0.344 − 1.95i)17-s + 4.70·18-s + (4.08 − 1.51i)19-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (1.22 − 1.03i)3-s + (−0.469 + 0.171i)4-s + (0.868 + 0.728i)6-s + (0.114 − 0.198i)7-s + (−0.176 − 0.306i)8-s + (0.272 − 1.54i)9-s + (−0.348 − 0.603i)11-s + (−0.400 + 0.694i)12-s + (−1.40 − 1.18i)13-s + (0.152 + 0.0555i)14-s + (0.191 − 0.160i)16-s + (−0.0834 − 0.473i)17-s + 1.10·18-s + (0.937 − 0.347i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.518 + 0.855i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.518 + 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90336 - 1.07216i\)
\(L(\frac12)\) \(\approx\) \(1.90336 - 1.07216i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.08 + 1.51i)T \)
good3 \( 1 + (-2.12 + 1.78i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-0.303 + 0.526i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.15 + 2.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.07 + 4.25i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.344 + 1.95i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-4.39 + 1.59i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.744 - 4.22i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.24 + 3.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.721T + 37T^{2} \)
41 \( 1 + (9.15 - 7.68i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-11.0 - 4.01i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.863 + 4.89i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-5.28 + 1.92i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.438 - 2.48i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (4.58 - 1.66i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.802 - 4.55i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (5.66 + 2.06i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.771 - 0.647i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (2.36 - 1.98i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-7.56 + 13.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.62 - 7.23i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.38 + 7.86i)T + (-91.1 + 33.1i)T^{2} \)
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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

−9.550704738351142822362736223824, −8.836949463272053221271561802706, −7.958659947036791787615397380652, −7.48919522291818961918465803489, −6.88121875543996233457561030057, −5.64512386420127516315009838514, −4.73838943219790069817338442885, −3.18010421654963613889215919763, −2.64311063595774572165329439835, −0.861219843934242890944913594328, 1.97612893106711436673066740402, 2.76616485553391662844196327952, 3.82100396471850189971004733632, 4.60245758580749620447704389867, 5.35403550007080813765399919491, 7.04458957095133430819721636480, 7.87789895334270630502729779736, 8.930800417153492932681461144602, 9.377606364626252373123429441119, 10.05173294677206167119961028434

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