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Properties

Label 2-950-95.44-c1-0-19
Degree $2$
Conductor $950$
Sign $-0.159 + 0.987i$
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  + (−0.642 − 0.766i)2-s + (−0.613 + 1.68i)3-s + (−0.173 + 0.984i)4-s + (1.68 − 0.613i)6-s + (−1.17 − 0.680i)7-s + (0.866 − 0.500i)8-s + (−0.169 − 0.142i)9-s + (−3.22 − 5.59i)11-s + (−1.55 − 0.897i)12-s + (2.01 + 5.52i)13-s + (0.236 + 1.34i)14-s + (−0.939 − 0.342i)16-s + (−1.64 − 1.96i)17-s + 0.221i·18-s + (3.83 − 2.06i)19-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (−0.354 + 0.973i)3-s + (−0.0868 + 0.492i)4-s + (0.688 − 0.250i)6-s + (−0.445 − 0.257i)7-s + (0.306 − 0.176i)8-s + (−0.0566 − 0.0474i)9-s + (−0.973 − 1.68i)11-s + (−0.448 − 0.259i)12-s + (0.557 + 1.53i)13-s + (0.0631 + 0.358i)14-s + (−0.234 − 0.0855i)16-s + (−0.398 − 0.475i)17-s + 0.0522i·18-s + (0.880 − 0.473i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.159 + 0.987i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.358567 - 0.420942i\)
\(L(\frac12)\) \(\approx\) \(0.358567 - 0.420942i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 + 0.766i)T \)
5 \( 1 \)
19 \( 1 + (-3.83 + 2.06i)T \)
good3 \( 1 + (0.613 - 1.68i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (1.17 + 0.680i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.22 + 5.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.01 - 5.52i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.64 + 1.96i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (8.29 + 1.46i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (3.41 + 2.86i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.701 + 1.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.42iT - 37T^{2} \)
41 \( 1 + (-5.15 - 1.87i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-6.74 + 1.19i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-6.11 + 7.29i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (4.35 + 0.768i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-7.87 + 6.60i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.12 + 6.36i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-4.31 + 5.14i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.336 - 1.90i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-2.14 + 5.90i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (3.37 + 1.22i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (11.6 + 6.74i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.78 - 0.648i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (0.0793 + 0.0945i)T + (-16.8 + 95.5i)T^{2} \)
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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

−9.803231128180307730779596764984, −9.266492985614851318849592802053, −8.383285780377735571218087165347, −7.43894757819513598750132756314, −6.26578810811797935353602967369, −5.37023589309303814955462138623, −4.19712283733385955428171312571, −3.56334320330446113111693404626, −2.25177189510503165604874995017, −0.33022647766575309887513621516, 1.31984289006762836032268208328, 2.55668017911286956371771397790, 4.14178230701142418133643745499, 5.55395120414822904995291677361, 5.96302640293669644255605599777, 7.09339296085324848056787221880, 7.64400924447976559094299397209, 8.240959967097326106061797486577, 9.585450316507550282945904498192, 10.06214857206473592154447147129

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