Properties

Label 2-950-19.17-c1-0-19
Degree $2$
Conductor $950$
Sign $0.997 + 0.0646i$
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 + (0.0864 + 0.0725i)3-s + (−0.939 − 0.342i)4-s + (−0.0864 + 0.0725i)6-s + (0.772 + 1.33i)7-s + (0.5 − 0.866i)8-s + (−0.518 − 2.94i)9-s + (0.653 − 1.13i)11-s + (−0.0564 − 0.0977i)12-s + (−0.437 + 0.367i)13-s + (−1.45 + 0.528i)14-s + (0.766 + 0.642i)16-s + (0.878 − 4.98i)17-s + 2.98·18-s + (−0.546 − 4.32i)19-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (0.0499 + 0.0418i)3-s + (−0.469 − 0.171i)4-s + (−0.0353 + 0.0296i)6-s + (0.291 + 0.505i)7-s + (0.176 − 0.306i)8-s + (−0.172 − 0.980i)9-s + (0.196 − 0.341i)11-s + (−0.0162 − 0.0282i)12-s + (−0.121 + 0.101i)13-s + (−0.387 + 0.141i)14-s + (0.191 + 0.160i)16-s + (0.212 − 1.20i)17-s + 0.704·18-s + (−0.125 − 0.992i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.997 + 0.0646i$
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: \((2,\ 950,\ (\ :1/2),\ 0.997 + 0.0646i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38571 - 0.0448684i\)
\(L(\frac12)\) \(\approx\) \(1.38571 - 0.0448684i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.546 + 4.32i)T \)
good3 \( 1 + (-0.0864 - 0.0725i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (-0.772 - 1.33i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.653 + 1.13i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.437 - 0.367i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.878 + 4.98i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-4.86 - 1.77i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.660 - 3.74i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.49 + 2.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.31T + 37T^{2} \)
41 \( 1 + (8.57 + 7.19i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-8.85 + 3.22i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.368 - 2.08i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-7.19 - 2.61i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.0463 - 0.262i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-3.05 - 1.11i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.57 - 8.94i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-15.2 + 5.55i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-2.14 - 1.80i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-3.19 - 2.68i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (4.63 + 8.03i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.42 - 3.71i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.500 + 2.83i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.665333401056833349150525347504, −9.060487033226797606582469973999, −8.558963012771410700432098642115, −7.28853109713964963193442976433, −6.79738564557775803370913403551, −5.67665706979671530505378191121, −5.01468598406740311264414889727, −3.79413926382068384683753345648, −2.64323485400398373367669180936, −0.76660171342225982767979679006, 1.33507537437881415919259640970, 2.42559585658560592672647515492, 3.70745273351438391358889196467, 4.58286205501125597834392581664, 5.53390005249543646935917868291, 6.71660749861248121066628435186, 7.87519340654356797316356101625, 8.247475856536537166286779058641, 9.365475073580786075001033222645, 10.29044778147728251655223401057

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