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Properties

Label 2-950-19.9-c1-0-0
Degree $2$
Conductor $950$
Sign $-0.431 - 0.902i$
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 + (−0.673 + 0.565i)3-s + (−0.939 + 0.342i)4-s + (0.673 + 0.565i)6-s + (0.266 − 0.460i)7-s + (0.5 + 0.866i)8-s + (−0.386 + 2.19i)9-s + (0.5 + 0.866i)11-s + (0.439 − 0.761i)12-s + (−0.5 − 0.419i)13-s + (−0.5 − 0.181i)14-s + (0.766 − 0.642i)16-s + (−0.673 − 3.82i)17-s + 2.22·18-s + (−4.21 + 1.10i)19-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−0.388 + 0.326i)3-s + (−0.469 + 0.171i)4-s + (0.275 + 0.230i)6-s + (0.100 − 0.174i)7-s + (0.176 + 0.306i)8-s + (−0.128 + 0.730i)9-s + (0.150 + 0.261i)11-s + (0.126 − 0.219i)12-s + (−0.138 − 0.116i)13-s + (−0.133 − 0.0486i)14-s + (0.191 − 0.160i)16-s + (−0.163 − 0.926i)17-s + 0.524·18-s + (−0.967 + 0.253i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.431 - 0.902i$
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: \((2,\ 950,\ (\ :1/2),\ -0.431 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.219137 + 0.347792i\)
\(L(\frac12)\) \(\approx\) \(0.219137 + 0.347792i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.21 - 1.10i)T \)
good3 \( 1 + (0.673 - 0.565i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-0.266 + 0.460i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.419i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.673 + 3.82i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (4.25 - 1.55i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.773 - 4.38i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.37 - 2.38i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.958T + 37T^{2} \)
41 \( 1 + (4.78 - 4.01i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (4.70 + 1.71i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.134 - 0.761i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (3.62 - 1.31i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.08 - 6.17i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (4.62 - 1.68i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.233 - 1.32i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (5.33 + 1.94i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (5.72 - 4.80i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (0.543 - 0.455i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (5.28 - 9.15i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.75 - 3.15i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.0175 + 0.0994i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.39064943666104302770019823480, −9.718621446952052458732770005620, −8.768224157387537145812929845164, −7.929032194550718222811344760553, −7.00864425107617818416630821407, −5.78799390873111535401666065075, −4.87487917966366282707713998532, −4.13606692260187767326310383859, −2.85543051393648254979955947860, −1.69141074002638302831368510357, 0.20445843364861855879019466836, 1.90930456979630832457809571251, 3.59531618777238726149624448908, 4.54398587251902455482778070930, 5.79944985786840821865736582590, 6.27859483679115855675180545407, 7.07226309339802114154337522551, 8.155265555666867658367741422778, 8.742261516630900958750729903875, 9.657371497765296251153163967778

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