Properties

Label 2-950-19.4-c1-0-1
Degree $2$
Conductor $950$
Sign $0.752 - 0.658i$
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 + (−0.552 − 3.13i)3-s + (0.766 + 0.642i)4-s + (−0.552 + 3.13i)6-s + (−1.67 + 2.89i)7-s + (−0.500 − 0.866i)8-s + (−6.68 + 2.43i)9-s + (−3.09 − 5.35i)11-s + (1.58 − 2.75i)12-s + (0.128 − 0.727i)13-s + (2.56 − 2.15i)14-s + (0.173 + 0.984i)16-s + (0.815 + 0.296i)17-s + 7.11·18-s + (−2.59 + 3.49i)19-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.318 − 1.80i)3-s + (0.383 + 0.321i)4-s + (−0.225 + 1.27i)6-s + (−0.632 + 1.09i)7-s + (−0.176 − 0.306i)8-s + (−2.22 + 0.810i)9-s + (−0.932 − 1.61i)11-s + (0.458 − 0.794i)12-s + (0.0355 − 0.201i)13-s + (0.685 − 0.574i)14-s + (0.0434 + 0.246i)16-s + (0.197 + 0.0720i)17-s + 1.67·18-s + (−0.596 + 0.802i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.752 - 0.658i$
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: \((2,\ 950,\ (\ :1/2),\ 0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228771 + 0.0859040i\)
\(L(\frac12)\) \(\approx\) \(0.228771 + 0.0859040i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 3.49i)T \)
good3 \( 1 + (0.552 + 3.13i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (1.67 - 2.89i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.09 + 5.35i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.128 + 0.727i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.815 - 0.296i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-0.875 - 0.735i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-7.32 + 2.66i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.23 - 5.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.68T + 37T^{2} \)
41 \( 1 + (-1.68 - 9.54i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (1.42 - 1.19i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-0.311 + 0.113i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-5.44 - 4.56i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (4.56 + 1.66i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-5.53 - 4.64i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.74 + 1.00i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (5.27 - 4.42i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.35 + 7.69i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (0.399 + 2.26i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (5.72 - 9.91i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.404 + 2.29i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (6.43 + 2.34i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.27583137364991838635132322025, −8.941984482033014091011647170278, −8.326233590897712410076408354156, −7.85635702817228863730872322750, −6.69742138106273820796021995993, −6.04846002070118244947441330813, −5.46955996781839657663391019533, −3.11194169146032887688460421340, −2.52500933729249447796413816065, −1.18107622608205720230590842740, 0.15925055332147814217579215713, 2.56274211763499587632779247290, 3.85467473751730982116960791560, 4.61950549647394237544448487959, 5.39625170542725420528183272123, 6.65700337799226005527261557307, 7.37042060659115710866063175218, 8.560189746448606387078895076540, 9.404506751364805007553180768126, 10.03433679370647457206183218512

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