Properties

Label 2-950-95.37-c1-0-17
Degree $2$
Conductor $950$
Sign $0.158 + 0.987i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.267 − 0.267i)3-s − 1.00i·4-s + 0.378·6-s + (0.709 + 0.709i)7-s + (0.707 + 0.707i)8-s − 2.85i·9-s − 5.59·11-s + (−0.267 + 0.267i)12-s + (2.99 + 2.99i)13-s − 1.00·14-s − 1.00·16-s + (−2.20 − 2.20i)17-s + (2.02 + 2.02i)18-s + (3.84 − 2.05i)19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.154 − 0.154i)3-s − 0.500i·4-s + 0.154·6-s + (0.268 + 0.268i)7-s + (0.250 + 0.250i)8-s − 0.952i·9-s − 1.68·11-s + (−0.0772 + 0.0772i)12-s + (0.830 + 0.830i)13-s − 0.268·14-s − 0.250·16-s + (−0.534 − 0.534i)17-s + (0.476 + 0.476i)18-s + (0.881 − 0.472i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.158 + 0.987i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.158 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.562952 - 0.479643i\)
\(L(\frac12)\) \(\approx\) \(0.562952 - 0.479643i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
19 \( 1 + (-3.84 + 2.05i)T \)
good3 \( 1 + (0.267 + 0.267i)T + 3iT^{2} \)
7 \( 1 + (-0.709 - 0.709i)T + 7iT^{2} \)
11 \( 1 + 5.59T + 11T^{2} \)
13 \( 1 + (-2.99 - 2.99i)T + 13iT^{2} \)
17 \( 1 + (2.20 + 2.20i)T + 17iT^{2} \)
23 \( 1 + (-5.89 + 5.89i)T - 23iT^{2} \)
29 \( 1 + 9.34T + 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (6.61 - 6.61i)T - 37iT^{2} \)
41 \( 1 + 9.14iT - 41T^{2} \)
43 \( 1 + (0.927 - 0.927i)T - 43iT^{2} \)
47 \( 1 + (4.20 + 4.20i)T + 47iT^{2} \)
53 \( 1 + (2.44 + 2.44i)T + 53iT^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (-5.72 + 5.72i)T - 67iT^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + (-4.33 + 4.33i)T - 73iT^{2} \)
79 \( 1 + 5.47T + 79T^{2} \)
83 \( 1 + (-5.53 + 5.53i)T - 83iT^{2} \)
89 \( 1 - 8.99T + 89T^{2} \)
97 \( 1 + (9.16 - 9.16i)T - 97iT^{2} \)
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

−9.634077590788776436792451907934, −8.958257772241875228910820560609, −8.269289863203244324552402020047, −7.19954386867198029974689357025, −6.65600241621503907294698227984, −5.54000325611831912010036916178, −4.87217315616961836620555519480, −3.40979782530706564936782168959, −2.07950966926710495108579327216, −0.42294355086186487124671069120, 1.45899605891159062198919969449, 2.74529179350080741089387209133, 3.74700375402721646583001075222, 5.14378886250939101982920670071, 5.58710736242989966418799386559, 7.23637133088764197064005527730, 7.83582330858517430775905404252, 8.460076328032788341256082300084, 9.573924240987915646272303430150, 10.38490617838253632345342788889

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