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Properties

Label 2-950-19.4-c1-0-21
Degree $2$
Conductor $950$
Sign $0.682 - 0.730i$
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  + (0.939 + 0.342i)2-s + (0.326 + 1.85i)3-s + (0.766 + 0.642i)4-s + (−0.326 + 1.85i)6-s + (2.53 − 4.38i)7-s + (0.500 + 0.866i)8-s + (−0.5 + 0.181i)9-s + (0.705 + 1.22i)11-s + (−0.939 + 1.62i)12-s + (0.226 − 1.28i)13-s + (3.87 − 3.25i)14-s + (0.173 + 0.984i)16-s + (2.24 + 0.817i)17-s − 0.532·18-s + (2.23 − 3.74i)19-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.188 + 1.06i)3-s + (0.383 + 0.321i)4-s + (−0.133 + 0.755i)6-s + (0.957 − 1.65i)7-s + (0.176 + 0.306i)8-s + (−0.166 + 0.0606i)9-s + (0.212 + 0.368i)11-s + (−0.271 + 0.469i)12-s + (0.0628 − 0.356i)13-s + (1.03 − 0.869i)14-s + (0.0434 + 0.246i)16-s + (0.544 + 0.198i)17-s − 0.125·18-s + (0.512 − 0.858i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.682 - 0.730i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.682 - 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.67539 + 1.16152i\)
\(L(\frac12)\) \(\approx\) \(2.67539 + 1.16152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 \)
19 \( 1 + (-2.23 + 3.74i)T \)
good3 \( 1 + (-0.326 - 1.85i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-2.53 + 4.38i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.705 - 1.22i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.226 + 1.28i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-2.24 - 0.817i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-2.34 - 1.96i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (7.94 - 2.89i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.184 - 0.320i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.82T + 37T^{2} \)
41 \( 1 + (0.266 + 1.50i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.581 + 0.487i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (9.59 - 3.49i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (1.28 + 1.07i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.673 + 0.245i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-7.47 - 6.27i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (1.31 - 0.480i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (4.87 - 4.09i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.791 - 4.49i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (0.389 + 2.20i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.99 - 3.45i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.84 - 10.4i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.43 + 0.524i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.22871903504265119398509541637, −9.505427340439359707779854843922, −8.404995094187609002178145862127, −7.38275866423403035163880160442, −6.99072330737920964216619425042, −5.40320499358113165801540106616, −4.76331278553620797935714873842, −3.97246792122371564204168769430, −3.30818142132727712254559424787, −1.41840226673860812109187351595, 1.52069920173857174708051132479, 2.20931660270999447505758418357, 3.37149813973122150062101792313, 4.80248823127485119970290600039, 5.62598270356882530510914679191, 6.32948714500951951026836052251, 7.43080504905845906592936938508, 8.150315177374746459819668084405, 8.939085363878773794812282562598, 9.920657013121076502327766221556

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