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Properties

Label 2-950-95.18-c1-0-26
Degree $2$
Conductor $950$
Sign $-0.768 + 0.639i$
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  + (−0.707 − 0.707i)2-s + (2.20 − 2.20i)3-s + 1.00i·4-s − 3.12·6-s + (−1.65 + 1.65i)7-s + (0.707 − 0.707i)8-s − 6.74i·9-s − 0.804·11-s + (2.20 + 2.20i)12-s + (3.05 − 3.05i)13-s + 2.34·14-s − 1.00·16-s + (4.37 − 4.37i)17-s + (−4.76 + 4.76i)18-s + (3.48 + 2.62i)19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (1.27 − 1.27i)3-s + 0.500i·4-s − 1.27·6-s + (−0.626 + 0.626i)7-s + (0.250 − 0.250i)8-s − 2.24i·9-s − 0.242·11-s + (0.637 + 0.637i)12-s + (0.846 − 0.846i)13-s + 0.626·14-s − 0.250·16-s + (1.06 − 1.06i)17-s + (−1.12 + 1.12i)18-s + (0.799 + 0.601i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.768 + 0.639i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.768 + 0.639i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.574955 - 1.58999i\)
\(L(\frac12)\) \(\approx\) \(0.574955 - 1.58999i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
19 \( 1 + (-3.48 - 2.62i)T \)
good3 \( 1 + (-2.20 + 2.20i)T - 3iT^{2} \)
7 \( 1 + (1.65 - 1.65i)T - 7iT^{2} \)
11 \( 1 + 0.804T + 11T^{2} \)
13 \( 1 + (-3.05 + 3.05i)T - 13iT^{2} \)
17 \( 1 + (-4.37 + 4.37i)T - 17iT^{2} \)
23 \( 1 + (6.20 + 6.20i)T + 23iT^{2} \)
29 \( 1 + 5.64T + 29T^{2} \)
31 \( 1 + 3.12iT - 31T^{2} \)
37 \( 1 + (0.524 + 0.524i)T + 37iT^{2} \)
41 \( 1 - 1.45iT - 41T^{2} \)
43 \( 1 + (1.90 + 1.90i)T + 43iT^{2} \)
47 \( 1 + (0.499 - 0.499i)T - 47iT^{2} \)
53 \( 1 + (3.40 - 3.40i)T - 53iT^{2} \)
59 \( 1 - 2.77T + 59T^{2} \)
61 \( 1 - 6.17T + 61T^{2} \)
67 \( 1 + (0.345 + 0.345i)T + 67iT^{2} \)
71 \( 1 - 6.68iT - 71T^{2} \)
73 \( 1 + (1.05 + 1.05i)T + 73iT^{2} \)
79 \( 1 - 7.16T + 79T^{2} \)
83 \( 1 + (-4.12 - 4.12i)T + 83iT^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + (-8.25 - 8.25i)T + 97iT^{2} \)
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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

−9.539594850396708970912487236776, −8.839021193302150153711295893318, −7.957174311090439182785015361596, −7.65847092735473078648400999394, −6.50735560839201804655689081093, −5.63687939814755207194368905694, −3.66894657747212756320709933877, −2.98884970573500523599982763075, −2.13304289276250606613311022425, −0.819748258420084576322005609792, 1.80327001530690110109026381774, 3.45582422025295454238374364606, 3.79727038425619301535718712769, 5.05346693343386567978199871076, 6.08817797257427337768916384977, 7.33134525851503868609493614009, 8.004345483523116980781240231802, 8.800301087384446480738067325884, 9.557096102334747665306617428452, 10.00100468561042761479607179795

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