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Properties

Label 2-950-19.17-c1-0-1
Degree $2$
Conductor $950$
Sign $-0.926 - 0.376i$
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 + (−1.16 − 2.01i)7-s + (0.5 − 0.866i)8-s + (−0.0339 − 0.192i)9-s + (1.54 − 2.68i)11-s + (0.837 + 1.45i)12-s + (−5.18 + 4.35i)13-s + (2.18 − 0.794i)14-s + (0.766 + 0.642i)16-s + (0.818 − 4.64i)17-s + 0.195·18-s + (−0.846 + 4.27i)19-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (−0.740 − 0.621i)3-s + (−0.469 − 0.171i)4-s + (0.523 − 0.439i)6-s + (−0.439 − 0.760i)7-s + (0.176 − 0.306i)8-s + (−0.0113 − 0.0641i)9-s + (0.466 − 0.808i)11-s + (0.241 + 0.418i)12-s + (−1.43 + 1.20i)13-s + (0.583 − 0.212i)14-s + (0.191 + 0.160i)16-s + (0.198 − 1.12i)17-s + 0.0460·18-s + (−0.194 + 0.980i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.926 - 0.376i$
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: \((2,\ 950,\ (\ :1/2),\ -0.926 - 0.376i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0172489 + 0.0883220i\)
\(L(\frac12)\) \(\approx\) \(0.0172489 + 0.0883220i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.846 - 4.27i)T \)
good3 \( 1 + (1.28 + 1.07i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (1.16 + 2.01i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.54 + 2.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.18 - 4.35i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.818 + 4.64i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-2.99 - 1.09i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.115 - 0.653i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.25 - 3.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.57T + 37T^{2} \)
41 \( 1 + (-1.86 - 1.56i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.05 - 0.748i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.08 - 6.18i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (9.43 + 3.43i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.77 - 10.0i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (3.49 + 1.27i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.920 - 5.22i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (11.0 - 4.03i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-1.96 - 1.64i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (12.6 + 10.6i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-3.21 - 5.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.47 - 5.43i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.923 + 5.23i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.26853637974854880835099650486, −9.492013946415003252363700999240, −8.758413784005077728046173508084, −7.45140202664306044832817220249, −7.00222486327874784837634717764, −6.36372987217968899003032693665, −5.40366224497152731591152792973, −4.43171483950932697672348338362, −3.22483278947600253553899572727, −1.31524293851541730217508711017, 0.05073036581560833393961142199, 2.10183302526243273954525519194, 3.11796889357522341219508900027, 4.45812336198160203964915387486, 5.10781830501444660269361972565, 5.96027762433041352277386765706, 7.14111747490852960845655937477, 8.172173827693373950177848923114, 9.159600543535999406128979371876, 9.906406221256536900003158347287

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