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Properties

Label 2-750-125.94-c1-0-24
Degree $2$
Conductor $750$
Sign $-0.922 + 0.385i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.998 + 0.0627i)2-s + (0.904 − 0.425i)3-s + (0.992 − 0.125i)4-s + (−0.00451 − 2.23i)5-s + (−0.876 + 0.481i)6-s + (−1.58 − 2.18i)7-s + (−0.982 + 0.187i)8-s + (0.637 − 0.770i)9-s + (0.144 + 2.23i)10-s + (0.188 + 2.99i)11-s + (0.844 − 0.535i)12-s + (−5.06 − 4.19i)13-s + (1.72 + 2.08i)14-s + (−0.956 − 2.02i)15-s + (0.968 − 0.248i)16-s + (0.607 − 4.80i)17-s + ⋯
L(s)  = 1  + (−0.705 + 0.0443i)2-s + (0.522 − 0.245i)3-s + (0.496 − 0.0626i)4-s + (−0.00201 − 0.999i)5-s + (−0.357 + 0.196i)6-s + (−0.600 − 0.825i)7-s + (−0.347 + 0.0662i)8-s + (0.212 − 0.256i)9-s + (0.0458 + 0.705i)10-s + (0.0567 + 0.902i)11-s + (0.243 − 0.154i)12-s + (−1.40 − 1.16i)13-s + (0.460 + 0.556i)14-s + (−0.246 − 0.521i)15-s + (0.242 − 0.0621i)16-s + (0.147 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.922 + 0.385i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.922 + 0.385i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.137616 - 0.686608i\)
\(L(\frac12)\) \(\approx\) \(0.137616 - 0.686608i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.998 - 0.0627i)T \)
3 \( 1 + (-0.904 + 0.425i)T \)
5 \( 1 + (0.00451 + 2.23i)T \)
good7 \( 1 + (1.58 + 2.18i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (-0.188 - 2.99i)T + (-10.9 + 1.37i)T^{2} \)
13 \( 1 + (5.06 + 4.19i)T + (2.43 + 12.7i)T^{2} \)
17 \( 1 + (-0.607 + 4.80i)T + (-16.4 - 4.22i)T^{2} \)
19 \( 1 + (2.86 - 6.08i)T + (-12.1 - 14.6i)T^{2} \)
23 \( 1 + (-0.901 - 2.27i)T + (-16.7 + 15.7i)T^{2} \)
29 \( 1 + (-3.85 + 3.62i)T + (1.82 - 28.9i)T^{2} \)
31 \( 1 + (6.14 + 0.775i)T + (30.0 + 7.70i)T^{2} \)
37 \( 1 + (-2.52 - 9.83i)T + (-32.4 + 17.8i)T^{2} \)
41 \( 1 + (6.93 + 2.74i)T + (29.8 + 28.0i)T^{2} \)
43 \( 1 + (5.20 + 1.69i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (-10.0 - 1.91i)T + (43.6 + 17.3i)T^{2} \)
53 \( 1 + (-2.26 + 4.11i)T + (-28.3 - 44.7i)T^{2} \)
59 \( 1 + (-1.22 - 1.93i)T + (-25.1 + 53.3i)T^{2} \)
61 \( 1 + (1.38 - 0.548i)T + (44.4 - 41.7i)T^{2} \)
67 \( 1 + (-7.55 + 8.04i)T + (-4.20 - 66.8i)T^{2} \)
71 \( 1 + (-2.15 + 11.2i)T + (-66.0 - 26.1i)T^{2} \)
73 \( 1 + (9.71 + 6.16i)T + (31.0 + 66.0i)T^{2} \)
79 \( 1 + (3.40 + 7.24i)T + (-50.3 + 60.8i)T^{2} \)
83 \( 1 + (3.52 + 1.65i)T + (52.9 + 63.9i)T^{2} \)
89 \( 1 + (-0.633 + 0.998i)T + (-37.8 - 80.5i)T^{2} \)
97 \( 1 + (6.10 + 6.50i)T + (-6.09 + 96.8i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.977783683345850547247180162210, −9.240779556963612631239867426638, −8.150063302155972041157306106071, −7.57600513632612351661275316907, −6.85512839895886439290769477178, −5.50500515266319785220020021943, −4.47494619030277268732460127557, −3.19488609032537708980969254859, −1.86142119904901178648575372830, −0.39220264115690265508447542422, 2.22267749373109665220070050412, 2.86434777663683889325017910165, 4.06553872108203386526940803021, 5.59413428036811651215131457241, 6.67838365899342499734969164096, 7.15876419529023507656919160106, 8.450045155838796772602024403439, 9.017552676599275109980478332797, 9.762031568789148233033474162273, 10.62786085609376433725412987899

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