↑

Properties

Label 2-950-25.11-c1-0-11
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: $\chi_{950} (761, \cdot )$
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

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

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