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Properties

Label 2-950-95.64-c1-0-3
Degree $2$
Conductor $950$
Sign $-0.602 - 0.798i$
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  + (0.866 − 0.5i)2-s + (−0.410 + 0.236i)3-s + (0.499 − 0.866i)4-s + (−0.236 + 0.410i)6-s + 2.19i·7-s − 0.999i·8-s + (−1.38 + 2.40i)9-s − 4.96·11-s + 0.473i·12-s + (−1.97 − 1.14i)13-s + (1.09 + 1.89i)14-s + (−0.5 − 0.866i)16-s + (−5.86 + 3.38i)17-s + 2.77i·18-s + (3.12 + 3.03i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.236 + 0.136i)3-s + (0.249 − 0.433i)4-s + (−0.0967 + 0.167i)6-s + 0.828i·7-s − 0.353i·8-s + (−0.462 + 0.801i)9-s − 1.49·11-s + 0.136i·12-s + (−0.548 − 0.316i)13-s + (0.292 + 0.507i)14-s + (−0.125 − 0.216i)16-s + (−1.42 + 0.821i)17-s + 0.654i·18-s + (0.716 + 0.697i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.602 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.602 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.602 - 0.798i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.602 - 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.318018 + 0.638321i\)
\(L(\frac12)\) \(\approx\) \(0.318018 + 0.638321i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
19 \( 1 + (-3.12 - 3.03i)T \)
good3 \( 1 + (0.410 - 0.236i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 2.19iT - 7T^{2} \)
11 \( 1 + 4.96T + 11T^{2} \)
13 \( 1 + (1.97 + 1.14i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.86 - 3.38i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (6.65 + 3.84i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.983 + 1.70i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.58T + 31T^{2} \)
37 \( 1 - 7.38iT - 37T^{2} \)
41 \( 1 + (5.70 + 9.87i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.79 - 5.07i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.62 - 2.09i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.62 - 2.09i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.72 - 2.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.36 + 4.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.700 - 0.404i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.59 - 9.69i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.20 - 2.42i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.63 - 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.7iT - 83T^{2} \)
89 \( 1 + (1.78 - 3.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.149 + 0.0861i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.33860850738388483081101401456, −9.931038207094542137167190375389, −8.352520232665214999834033547048, −8.138164531689475351053734193503, −6.69666966143891208468936615725, −5.70875484274753695777124815367, −5.17552269566145670521033387878, −4.24789633140777122771573825099, −2.73851778512652019521704507012, −2.19114671704699852432850642621, 0.24988469118070087061733729794, 2.36835567274701515608560255620, 3.41774803514027115681511470313, 4.59660754609920573132274685808, 5.27555550551639491320223006374, 6.36780952895655601049031325264, 7.10483279671402338138872116800, 7.79625256291824020755507019783, 8.825999460812411606281004279783, 9.824313461781235821455288311052

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