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Properties

Label 2-950-95.18-c1-0-2
Degree $2$
Conductor $950$
Sign $-0.970 + 0.242i$
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 + (−0.707 + 0.707i)3-s + 1.00i·4-s + 1.00·6-s + (−2.73 + 2.73i)7-s + (0.707 − 0.707i)8-s + 1.99i·9-s + (−0.707 − 0.707i)12-s + (0.707 − 0.707i)13-s + 3.87·14-s − 1.00·16-s + (−2.73 + 2.73i)17-s + (1.41 − 1.41i)18-s + (3.87 − 2i)19-s − 3.87i·21-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + 0.408·6-s + (−1.03 + 1.03i)7-s + (0.250 − 0.250i)8-s + 0.666i·9-s + (−0.204 − 0.204i)12-s + (0.196 − 0.196i)13-s + 1.03·14-s − 0.250·16-s + (−0.664 + 0.664i)17-s + (0.333 − 0.333i)18-s + (0.888 − 0.458i)19-s − 0.845i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.970 + 0.242i$
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: \((2,\ 950,\ (\ :1/2),\ -0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0119839 - 0.0973964i\)
\(L(\frac12)\) \(\approx\) \(0.0119839 - 0.0973964i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.87 + 2i)T \)
good3 \( 1 + (0.707 - 0.707i)T - 3iT^{2} \)
7 \( 1 + (2.73 - 2.73i)T - 7iT^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - 13iT^{2} \)
17 \( 1 + (2.73 - 2.73i)T - 17iT^{2} \)
23 \( 1 + (2.73 + 2.73i)T + 23iT^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (1.41 + 1.41i)T + 37iT^{2} \)
41 \( 1 + 7.74iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (5.47 - 5.47i)T - 47iT^{2} \)
53 \( 1 + (-6.36 + 6.36i)T - 53iT^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + (9.19 + 9.19i)T + 67iT^{2} \)
71 \( 1 - 7.74iT - 71T^{2} \)
73 \( 1 + (-2.73 - 2.73i)T + 73iT^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + (5.47 + 5.47i)T + 83iT^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + (-5.65 - 5.65i)T + 97iT^{2} \)
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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

−10.42607445997929508895003175247, −9.727443879939420005938503757284, −8.989342024613477425491608316635, −8.246583728353543150033585655533, −7.15956562270353590576873376997, −6.10443564340766711160076660868, −5.35464291009671134161923141984, −4.16407150588410882623457979040, −3.03393804118443922509107563096, −2.03542878904362316448875576227, 0.05910300256377341800175180493, 1.33319641656761776765808620067, 3.19932712350055463798291167771, 4.19910252736097675682541697822, 5.56769998812837264518494067449, 6.37307630473771322607380554748, 7.04207465147411283907940076130, 7.61628885707309517484394586675, 8.846072360569994291276141313488, 9.659350978132817380853251309348

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