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Properties

Label 2-950-95.33-c1-0-27
Degree $2$
Conductor $950$
Sign $-0.141 + 0.989i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.141 + 0.989i$
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: \((2,\ 950,\ (\ :1/2),\ -0.141 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.840907 - 0.969485i\)
\(L(\frac12)\) \(\approx\) \(0.840907 - 0.969485i\)
\(L(\frac{3}{2})\) 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

−9.765693754512389902249461604171, −9.114306955088370371044196412408, −7.84584427911776506714891660850, −7.12055855918885257924152060277, −6.28827022840251045850683418364, −5.69356825949216183877368860286, −4.37311542529394948928290657907, −3.52695001188387297535155801733, −2.55945786950872097052102169648, −0.45860931836515071073392167547, 1.95540064154849127573350126536, 2.84037615091398007420565066670, 4.20357242656026713199771441370, 4.79714651734726246128298422803, 5.86932105671155983594565257470, 6.89387892104024911103390648429, 7.35808065046747149323508081460, 8.886263540188911384702968016533, 9.611460719222809019512590948502, 10.24375236089025858069171090409

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