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Properties

Label 2-950-19.17-c1-0-2
Degree $2$
Conductor $950$
Sign $0.427 - 0.904i$
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 + (−1.23 − 2.14i)7-s + (−0.5 + 0.866i)8-s + (−0.0320 − 0.181i)9-s + (−1.45 + 2.51i)11-s + (0.838 + 1.45i)12-s + (−0.993 + 0.833i)13-s + (−2.32 + 0.846i)14-s + (0.766 + 0.642i)16-s + (−0.215 + 1.22i)17-s − 0.184·18-s + (−4.22 − 1.08i)19-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.742 − 0.622i)3-s + (−0.469 − 0.171i)4-s + (−0.524 + 0.440i)6-s + (−0.467 − 0.810i)7-s + (−0.176 + 0.306i)8-s + (−0.0106 − 0.0606i)9-s + (−0.437 + 0.757i)11-s + (0.242 + 0.419i)12-s + (−0.275 + 0.231i)13-s + (−0.621 + 0.226i)14-s + (0.191 + 0.160i)16-s + (−0.0523 + 0.296i)17-s − 0.0435·18-s + (−0.968 − 0.248i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.427 - 0.904i$
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: \((2,\ 950,\ (\ :1/2),\ 0.427 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0992122 + 0.0628275i\)
\(L(\frac12)\) \(\approx\) \(0.0992122 + 0.0628275i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.22 + 1.08i)T \)
good3 \( 1 + (1.28 + 1.07i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (1.23 + 2.14i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.45 - 2.51i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.993 - 0.833i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.215 - 1.22i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-2.06 - 0.750i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.12 + 6.38i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-3.04 - 5.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.44T + 37T^{2} \)
41 \( 1 + (-1.82 - 1.53i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (8.64 - 3.14i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.93 - 10.9i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-11.3 - 4.11i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.351 - 1.99i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-1.25 - 0.458i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.584 + 3.31i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (14.2 - 5.19i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (0.429 + 0.360i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-4.96 - 4.16i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (4.22 + 7.32i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.11 - 7.64i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.469 + 2.66i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.30789629296738191771087314915, −9.649753024593132772048588160585, −8.619080353642640551220187398199, −7.46077175397810891567536329502, −6.76475899800368650598493142459, −5.94752357665102105924693947465, −4.79324390151224373459512284357, −3.95096951134283614907720105438, −2.65478648695218604283153428050, −1.31621377041730030558603583137, 0.06042242042908705253974875515, 2.52319749633882281028809700624, 3.76762090568933491185268630757, 4.94319859256578110158132268250, 5.52813876550266932183018067339, 6.22713723221192018062613430663, 7.21075046865378102616797828130, 8.356101114134482282605800062955, 8.889407212878702265815753231671, 9.985530587938107700031765683204

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