↑

Properties

Label 2-950-95.12-c1-0-10
Degree $2$
Conductor $950$
Sign $0.994 + 0.106i$
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  + (−0.965 − 0.258i)2-s + (0.394 − 1.47i)3-s + (0.866 + 0.499i)4-s + (−0.761 + 1.31i)6-s + (1.69 + 1.69i)7-s + (−0.707 − 0.707i)8-s + (0.586 + 0.338i)9-s + 4.92·11-s + (1.07 − 1.07i)12-s + (−3.55 + 0.951i)13-s + (−1.19 − 2.07i)14-s + (0.500 + 0.866i)16-s + (−0.0410 + 0.153i)17-s + (−0.479 − 0.479i)18-s + (0.533 + 4.32i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.227 − 0.849i)3-s + (0.433 + 0.249i)4-s + (−0.311 + 0.538i)6-s + (0.640 + 0.640i)7-s + (−0.249 − 0.249i)8-s + (0.195 + 0.112i)9-s + 1.48·11-s + (0.311 − 0.311i)12-s + (−0.985 + 0.263i)13-s + (−0.320 − 0.555i)14-s + (0.125 + 0.216i)16-s + (−0.00995 + 0.0371i)17-s + (−0.112 − 0.112i)18-s + (0.122 + 0.992i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.994 + 0.106i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.994 + 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43217 - 0.0762951i\)
\(L(\frac12)\) \(\approx\) \(1.43217 - 0.0762951i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
19 \( 1 + (-0.533 - 4.32i)T \)
good3 \( 1 + (-0.394 + 1.47i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.69 - 1.69i)T + 7iT^{2} \)
11 \( 1 - 4.92T + 11T^{2} \)
13 \( 1 + (3.55 - 0.951i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.0410 - 0.153i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-0.620 - 2.31i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.62 - 8.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.10iT - 31T^{2} \)
37 \( 1 + (2.44 - 2.44i)T - 37iT^{2} \)
41 \( 1 + (-6.51 + 3.75i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.15 - 1.64i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.62 - 0.702i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.03 + 1.61i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.77 + 3.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.36 + 4.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.36 + 8.84i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.64 + 2.10i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.16 + 0.847i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.39 + 12.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.27 - 2.27i)T - 83iT^{2} \)
89 \( 1 + (-3.93 + 6.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (16.8 + 4.52i)T + (84.0 + 48.5i)T^{2} \)
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

−9.862954216281940223972639628498, −9.069846498419317427991006560484, −8.441238160103588112187407889888, −7.44012472215282256563328362982, −7.00011447819405694319575750210, −5.93494896021694370190543787242, −4.74921101539675985457505068463, −3.43756591062036751150166925878, −2.02060547470413959952772313576, −1.40667921156072831686862492233, 0.944562816215612289544008752446, 2.48130902119923573810239721707, 4.00746089086990866164753659465, 4.49424413163291078501780867754, 5.77894375386203411410204466213, 6.94666927755497904080700265764, 7.49887049195718551173289996153, 8.568670837870665167259924728299, 9.470707247445347002144227398880, 9.713939088448877156901854884982

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