↑

Properties

Label 2-950-25.16-c1-0-34
Degree $2$
Conductor $950$
Sign $-0.183 + 0.983i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.207 − 0.639i)3-s + (0.309 + 0.951i)4-s + (2.21 − 0.285i)5-s + (−0.207 + 0.639i)6-s + 0.622·7-s + (0.309 − 0.951i)8-s + (2.06 − 1.49i)9-s + (−1.96 − 1.07i)10-s + (−3.88 − 2.82i)11-s + (0.543 − 0.395i)12-s + (3.28 − 2.38i)13-s + (−0.503 − 0.365i)14-s + (−0.642 − 1.35i)15-s + (−0.809 + 0.587i)16-s + (−0.839 + 2.58i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.119 − 0.369i)3-s + (0.154 + 0.475i)4-s + (0.991 − 0.127i)5-s + (−0.0848 + 0.261i)6-s + 0.235·7-s + (0.109 − 0.336i)8-s + (0.687 − 0.499i)9-s + (−0.620 − 0.339i)10-s + (−1.17 − 0.850i)11-s + (0.157 − 0.114i)12-s + (0.911 − 0.662i)13-s + (−0.134 − 0.0977i)14-s + (−0.166 − 0.350i)15-s + (−0.202 + 0.146i)16-s + (−0.203 + 0.626i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.183 + 0.983i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.183 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869789 - 1.04680i\)
\(L(\frac12)\) \(\approx\) \(0.869789 - 1.04680i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-2.21 + 0.285i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.207 + 0.639i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 0.622T + 7T^{2} \)
11 \( 1 + (3.88 + 2.82i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.28 + 2.38i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.839 - 2.58i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (2.84 + 2.06i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.480 + 1.48i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.745 - 2.29i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-8.95 + 6.50i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.09 + 2.25i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.47T + 43T^{2} \)
47 \( 1 + (-1.66 - 5.13i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.33 - 7.18i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.91 - 2.11i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (6.52 + 4.74i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.67 + 8.22i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.18 + 6.72i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.32 + 0.962i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.80 + 8.64i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.94 - 12.1i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (12.7 + 9.26i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.73 - 11.4i)T + (-78.4 + 57.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.865662730912370790740007651463, −9.029504338202901913141452538550, −8.196619456283757157385522903040, −7.51190317969310382436726384805, −6.15642980581865817237484875972, −5.84340704721268056744638652742, −4.37862759349221135382587118922, −3.11237894528032915571572875272, −1.97963627402605725524713765973, −0.823689070875369331630984911004, 1.56385510000420752710842519773, 2.54815374035232460669606222464, 4.34261240433508112634117412305, 5.13132583153380079594972612680, 5.99941032770476013517983229338, 6.98107925081366422286005700809, 7.70470943994154154318848757640, 8.678756224967537286947420956273, 9.708214679281900516242740463953, 9.956583401244440977557372041951

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