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Properties

Label 2-950-19.11-c1-0-21
Degree $2$
Conductor $950$
Sign $0.321 + 0.946i$
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  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.499 − 0.866i)6-s + 2·7-s − 0.999·8-s + (1 + 1.73i)9-s − 0.999·12-s + (3 + 5.19i)13-s + (1 − 1.73i)14-s + (−0.5 + 0.866i)16-s + (3.5 − 6.06i)17-s + 2·18-s + (3.5 − 2.59i)19-s + (1 − 1.73i)21-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.204 − 0.353i)6-s + 0.755·7-s − 0.353·8-s + (0.333 + 0.577i)9-s − 0.288·12-s + (0.832 + 1.44i)13-s + (0.267 − 0.462i)14-s + (−0.125 + 0.216i)16-s + (0.848 − 1.47i)17-s + 0.471·18-s + (0.802 − 0.596i)19-s + (0.218 − 0.377i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.321 + 0.946i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.321 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94497 - 1.39292i\)
\(L(\frac12)\) \(\approx\) \(1.94497 - 1.39292i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
19 \( 1 + (-3.5 + 2.59i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5 + 8.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6 - 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6 + 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.5 - 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13T + 83T^{2} \)
89 \( 1 + (-6.5 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.5 + 12.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.753896958038120181784811519343, −9.257148210482455452088647790531, −8.119250635477307103428660859525, −7.43396729172931931625275618412, −6.50209472958931857661344709605, −5.23211496517706792481320258497, −4.58659887787054676319870682825, −3.39247267870296325394236208298, −2.17986438884057206823037595684, −1.27015546854278142233877311079, 1.39439241757736951094852567552, 3.40874993968444127245205612980, 3.73999173967527726462976642243, 5.15097657793165521358766752593, 5.68040130349886059775860832322, 6.79156902407378647287619746137, 7.82417661959673157211801893168, 8.395764872650266971956110983382, 9.180718358012264093291878714398, 10.34599039577363938155291269900

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