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Properties

Label 2-950-95.8-c1-0-11
Degree $2$
Conductor $950$
Sign $0.832 - 0.553i$
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  + (0.965 − 0.258i)2-s + (0.313 + 1.17i)3-s + (0.866 − 0.499i)4-s + (0.606 + 1.04i)6-s + (−1.88 + 1.88i)7-s + (0.707 − 0.707i)8-s + (1.32 − 0.765i)9-s + 5.08·11-s + (0.857 + 0.857i)12-s + (1.59 + 0.428i)13-s + (−1.33 + 2.30i)14-s + (0.500 − 0.866i)16-s + (−2.07 − 7.73i)17-s + (1.08 − 1.08i)18-s + (1.52 + 4.08i)19-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.181 + 0.676i)3-s + (0.433 − 0.249i)4-s + (0.247 + 0.428i)6-s + (−0.712 + 0.712i)7-s + (0.249 − 0.249i)8-s + (0.441 − 0.255i)9-s + 1.53·11-s + (0.247 + 0.247i)12-s + (0.443 + 0.118i)13-s + (−0.356 + 0.617i)14-s + (0.125 − 0.216i)16-s + (−0.502 − 1.87i)17-s + (0.255 − 0.255i)18-s + (0.349 + 0.936i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.57450 + 0.777146i\)
\(L(\frac12)\) \(\approx\) \(2.57450 + 0.777146i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
19 \( 1 + (-1.52 - 4.08i)T \)
good3 \( 1 + (-0.313 - 1.17i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.88 - 1.88i)T - 7iT^{2} \)
11 \( 1 - 5.08T + 11T^{2} \)
13 \( 1 + (-1.59 - 0.428i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.07 + 7.73i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (2.16 - 8.09i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.858 + 1.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.07iT - 31T^{2} \)
37 \( 1 + (-0.618 - 0.618i)T + 37iT^{2} \)
41 \( 1 + (-4.93 - 2.84i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.23 + 1.13i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (9.97 + 2.67i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.22 - 1.66i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-7.18 + 12.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.82 + 4.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.363 + 1.35i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (13.2 + 7.63i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.99 - 0.535i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.01 + 3.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.12 + 3.12i)T + 83iT^{2} \)
89 \( 1 + (5.26 + 9.11i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.0 - 2.68i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.771449027477772801949828258534, −9.566491038277880628717737984134, −8.801031627574143826486780326496, −7.33198287461530032680596596394, −6.56250063934412248023967650548, −5.72230761989446650265453844535, −4.68705446486325577857858486422, −3.72511829533771317014742650042, −3.11295229155236037477481939477, −1.52266523090404825011078511666, 1.19632747859103598048684076052, 2.47834616441424445677579298516, 3.99717240092370072593037697991, 4.21904057181173940494869551781, 5.95881282668474371911910282257, 6.58048923054938620142426210478, 7.10902949618511211175681746496, 8.162489672444019784774312327229, 8.970315039573656764705913969935, 10.11009958679092976805059330157

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