↑

Properties

Label 2-750-125.91-c1-0-19
Degree $2$
Conductor $750$
Sign $-0.972 + 0.232i$
Analytic cond. $5.98878$
Root an. cond. $2.44719$
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  + (−0.728 − 0.684i)2-s + (−0.535 + 0.844i)3-s + (0.0627 + 0.998i)4-s + (0.595 − 2.15i)5-s + (0.968 − 0.248i)6-s + (−0.577 − 1.77i)7-s + (0.637 − 0.770i)8-s + (−0.425 − 0.904i)9-s + (−1.90 + 1.16i)10-s + (−0.509 − 0.478i)11-s + (−0.876 − 0.481i)12-s + (−0.546 − 1.16i)13-s + (−0.796 + 1.69i)14-s + (1.50 + 1.65i)15-s + (−0.992 + 0.125i)16-s + (−0.292 + 4.64i)17-s + ⋯
L(s)  = 1  + (−0.515 − 0.484i)2-s + (−0.309 + 0.487i)3-s + (0.0313 + 0.499i)4-s + (0.266 − 0.963i)5-s + (0.395 − 0.101i)6-s + (−0.218 − 0.672i)7-s + (0.225 − 0.272i)8-s + (−0.141 − 0.301i)9-s + (−0.603 + 0.367i)10-s + (−0.153 − 0.144i)11-s + (−0.252 − 0.139i)12-s + (−0.151 − 0.321i)13-s + (−0.212 + 0.452i)14-s + (0.387 + 0.428i)15-s + (−0.248 + 0.0313i)16-s + (−0.0709 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.972 + 0.232i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.972 + 0.232i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0578353 - 0.491536i\)
\(L(\frac12)\) \(\approx\) \(0.0578353 - 0.491536i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.728 + 0.684i)T \)
3 \( 1 + (0.535 - 0.844i)T \)
5 \( 1 + (-0.595 + 2.15i)T \)
good7 \( 1 + (0.577 + 1.77i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.509 + 0.478i)T + (0.690 + 10.9i)T^{2} \)
13 \( 1 + (0.546 + 1.16i)T + (-8.28 + 10.0i)T^{2} \)
17 \( 1 + (0.292 - 4.64i)T + (-16.8 - 2.13i)T^{2} \)
19 \( 1 + (4.43 + 6.98i)T + (-8.08 + 17.1i)T^{2} \)
23 \( 1 + (-1.29 - 6.77i)T + (-21.3 + 8.46i)T^{2} \)
29 \( 1 + (5.34 - 2.11i)T + (21.1 - 19.8i)T^{2} \)
31 \( 1 + (-0.176 + 2.80i)T + (-30.7 - 3.88i)T^{2} \)
37 \( 1 + (-3.65 + 0.461i)T + (35.8 - 9.20i)T^{2} \)
41 \( 1 + (-1.49 + 7.81i)T + (-38.1 - 15.0i)T^{2} \)
43 \( 1 + (8.50 - 6.18i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (6.39 + 7.73i)T + (-8.80 + 46.1i)T^{2} \)
53 \( 1 + (3.30 + 0.847i)T + (46.4 + 25.5i)T^{2} \)
59 \( 1 + (8.51 + 4.68i)T + (31.6 + 49.8i)T^{2} \)
61 \( 1 + (0.554 + 2.90i)T + (-56.7 + 22.4i)T^{2} \)
67 \( 1 + (14.9 + 5.90i)T + (48.8 + 45.8i)T^{2} \)
71 \( 1 + (-2.49 - 3.01i)T + (-13.3 + 69.7i)T^{2} \)
73 \( 1 + (-3.35 + 1.84i)T + (39.1 - 61.6i)T^{2} \)
79 \( 1 + (0.432 - 0.682i)T + (-33.6 - 71.4i)T^{2} \)
83 \( 1 + (-4.04 - 6.37i)T + (-35.3 + 75.1i)T^{2} \)
89 \( 1 + (2.60 - 1.43i)T + (47.6 - 75.1i)T^{2} \)
97 \( 1 + (-0.461 + 0.182i)T + (70.7 - 66.4i)T^{2} \)
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

−9.912014965021022422947841861393, −9.251808366169743864306681753224, −8.500025859542878879507312541196, −7.55382370974440749220255865434, −6.41797021211965656965240785083, −5.31060954472848843108617080483, −4.38012081286186323582861750988, −3.42300578393726883551204251791, −1.80356884702367357670013070164, −0.29758874431933399053921522144, 1.88591309564264706035884109167, 2.92365790398145600880315113679, 4.61259647746760989527101144677, 5.89207359394457833249246214133, 6.35337758441204433777746481048, 7.23476830816479206841571627317, 8.034539052510062572474455173149, 9.020329991441781413475743207030, 9.909537672379220878926193443403, 10.60906621482431294947019509924

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