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Properties

Label 2-950-95.74-c1-0-23
Degree $2$
Conductor $950$
Sign $-0.338 + 0.941i$
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  + (−0.984 − 0.173i)2-s + (1.43 − 1.71i)3-s + (0.939 + 0.342i)4-s + (−1.71 + 1.43i)6-s + (−2.43 + 1.40i)7-s + (−0.866 − 0.5i)8-s + (−0.345 − 1.96i)9-s + (3.20 − 5.54i)11-s + (1.93 − 1.11i)12-s + (3.07 + 3.65i)13-s + (2.64 − 0.963i)14-s + (0.766 + 0.642i)16-s + (−6.82 − 1.20i)17-s + 1.99i·18-s + (2.52 − 3.55i)19-s + ⋯
L(s)  = 1  + (−0.696 − 0.122i)2-s + (0.829 − 0.988i)3-s + (0.469 + 0.171i)4-s + (−0.698 + 0.586i)6-s + (−0.921 + 0.532i)7-s + (−0.306 − 0.176i)8-s + (−0.115 − 0.653i)9-s + (0.965 − 1.67i)11-s + (0.558 − 0.322i)12-s + (0.851 + 1.01i)13-s + (0.707 − 0.257i)14-s + (0.191 + 0.160i)16-s + (−1.65 − 0.291i)17-s + 0.469i·18-s + (0.579 − 0.814i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.338 + 0.941i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.338 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.759546 - 1.07996i\)
\(L(\frac12)\) \(\approx\) \(0.759546 - 1.07996i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 + 0.173i)T \)
5 \( 1 \)
19 \( 1 + (-2.52 + 3.55i)T \)
good3 \( 1 + (-1.43 + 1.71i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (2.43 - 1.40i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.20 + 5.54i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.07 - 3.65i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (6.82 + 1.20i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-1.86 + 5.11i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.30 + 7.41i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-0.656 - 1.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.63iT - 37T^{2} \)
41 \( 1 + (1.76 + 1.48i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.83 + 7.80i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-7.59 + 1.33i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (1.19 - 3.28i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.48 - 8.39i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-2.80 - 1.02i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (8.84 - 1.56i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.34 + 0.851i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (4.50 - 5.37i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-2.36 - 1.98i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-0.320 + 0.185i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.130 + 0.109i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-14.4 - 2.55i)T + (91.1 + 33.1i)T^{2} \)
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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

−9.180608304209151669642590993159, −8.945222188683086503914310994230, −8.494084738805208624600530581559, −7.19318650499095514716859120098, −6.59059150978544669036905131652, −5.98035788630144675967447934941, −4.06673142986591271771052166971, −2.96901764619982524032484520381, −2.15851212464530377401727830647, −0.72468653428194120957134617862, 1.56763838193044483279851086182, 3.13326990147698945284687933459, 3.81766120946172364916310482309, 4.82836529395514962472120028566, 6.30221398579848522107040723195, 6.98542860755235698907666803825, 7.952203343161070580842224631016, 8.937481587178382644632333805111, 9.441472679737047133838867540541, 10.05177221381045704129561889827

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