Properties

Label 2-950-95.33-c1-0-14
Degree $2$
Conductor $950$
Sign $0.941 + 0.336i$
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  + (−0.906 − 0.422i)2-s + (0.199 + 0.284i)3-s + (0.642 + 0.766i)4-s + (−0.0603 − 0.342i)6-s + (3.03 + 0.814i)7-s + (−0.258 − 0.965i)8-s + (0.984 − 2.70i)9-s + (1.15 − 2.00i)11-s + (−0.0898 + 0.335i)12-s + (5.26 + 3.68i)13-s + (−2.40 − 2.02i)14-s + (−0.173 + 0.984i)16-s + (0.101 − 0.217i)17-s + (−2.03 + 2.03i)18-s + (−4.33 + 0.449i)19-s + ⋯
L(s)  = 1  + (−0.640 − 0.298i)2-s + (0.115 + 0.164i)3-s + (0.321 + 0.383i)4-s + (−0.0246 − 0.139i)6-s + (1.14 + 0.307i)7-s + (−0.0915 − 0.341i)8-s + (0.328 − 0.901i)9-s + (0.349 − 0.604i)11-s + (−0.0259 + 0.0968i)12-s + (1.46 + 1.02i)13-s + (−0.643 − 0.540i)14-s + (−0.0434 + 0.246i)16-s + (0.0246 − 0.0527i)17-s + (−0.479 + 0.479i)18-s + (−0.994 + 0.103i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.941 + 0.336i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.941 + 0.336i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48358 - 0.257084i\)
\(L(\frac12)\) \(\approx\) \(1.48358 - 0.257084i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.906 + 0.422i)T \)
5 \( 1 \)
19 \( 1 + (4.33 - 0.449i)T \)
good3 \( 1 + (-0.199 - 0.284i)T + (-1.02 + 2.81i)T^{2} \)
7 \( 1 + (-3.03 - 0.814i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.15 + 2.00i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.26 - 3.68i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.101 + 0.217i)T + (-10.9 - 13.0i)T^{2} \)
23 \( 1 + (3.03 + 0.265i)T + (22.6 + 3.99i)T^{2} \)
29 \( 1 + (9.33 + 3.39i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-7.73 + 4.46i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.25 - 2.25i)T + 37iT^{2} \)
41 \( 1 + (-11.4 - 2.01i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.0171 + 0.196i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (3.38 - 1.58i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (0.0547 - 0.625i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (-9.92 + 3.61i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (7.07 - 5.93i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-4.90 - 10.5i)T + (-43.0 + 51.3i)T^{2} \)
71 \( 1 + (-1.44 + 1.72i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + (-12.5 + 8.79i)T + (24.9 - 68.5i)T^{2} \)
79 \( 1 + (-1.43 + 8.13i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.739 - 2.76i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-2.79 - 15.8i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (0.340 + 0.158i)T + (62.3 + 74.3i)T^{2} \)
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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

−9.832023042467776124755287578336, −9.113253600972694328574646294718, −8.467204893271502660702955129710, −7.80629645474972504056331350003, −6.48998606034309819525401999901, −5.96645048337176654961491962513, −4.32708789865459386451667333392, −3.76748104677781879936635454478, −2.20569187998131896437885308598, −1.11443948826052973790915225615, 1.24290818473129692959973507503, 2.19397826057375385699911310052, 3.88830495109039195120984223069, 4.89944079370673053311956016777, 5.85965904486327419358973298574, 6.88175099039188104599680786705, 7.900111367502821698080121776636, 8.135237951824174553784368763467, 9.085992028211302372085259146649, 10.18691426223706259764574950980

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