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Properties

Label 2-750-125.94-c1-0-1
Degree $2$
Conductor $750$
Sign $-0.868 - 0.496i$
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 + (−0.667 + 2.13i)5-s + (0.876 − 0.481i)6-s + (−0.730 − 1.00i)7-s + (−0.982 + 0.187i)8-s + (0.637 − 0.770i)9-s + (0.531 − 2.17i)10-s + (−0.175 − 2.78i)11-s + (−0.844 + 0.535i)12-s + (4.50 + 3.72i)13-s + (0.792 + 0.957i)14-s + (−0.304 − 2.21i)15-s + (0.968 − 0.248i)16-s + (−0.848 + 6.71i)17-s + ⋯
L(s)  = 1  + (−0.705 + 0.0443i)2-s + (−0.522 + 0.245i)3-s + (0.496 − 0.0626i)4-s + (−0.298 + 0.954i)5-s + (0.357 − 0.196i)6-s + (−0.276 − 0.380i)7-s + (−0.347 + 0.0662i)8-s + (0.212 − 0.256i)9-s + (0.168 − 0.686i)10-s + (−0.0528 − 0.839i)11-s + (−0.243 + 0.154i)12-s + (1.24 + 1.03i)13-s + (0.211 + 0.255i)14-s + (−0.0787 − 0.571i)15-s + (0.242 − 0.0621i)16-s + (−0.205 + 1.62i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.868 - 0.496i$
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: \((2,\ 750,\ (\ :1/2),\ -0.868 - 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123654 + 0.465251i\)
\(L(\frac12)\) \(\approx\) \(0.123654 + 0.465251i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.667 - 2.13i)T \)
good7 \( 1 + (0.730 + 1.00i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (0.175 + 2.78i)T + (-10.9 + 1.37i)T^{2} \)
13 \( 1 + (-4.50 - 3.72i)T + (2.43 + 12.7i)T^{2} \)
17 \( 1 + (0.848 - 6.71i)T + (-16.4 - 4.22i)T^{2} \)
19 \( 1 + (-0.281 + 0.598i)T + (-12.1 - 14.6i)T^{2} \)
23 \( 1 + (0.207 + 0.523i)T + (-16.7 + 15.7i)T^{2} \)
29 \( 1 + (4.94 - 4.64i)T + (1.82 - 28.9i)T^{2} \)
31 \( 1 + (-0.951 - 0.120i)T + (30.0 + 7.70i)T^{2} \)
37 \( 1 + (-0.819 - 3.19i)T + (-32.4 + 17.8i)T^{2} \)
41 \( 1 + (10.5 + 4.16i)T + (29.8 + 28.0i)T^{2} \)
43 \( 1 + (6.65 + 2.16i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (2.11 + 0.403i)T + (43.6 + 17.3i)T^{2} \)
53 \( 1 + (1.85 - 3.36i)T + (-28.3 - 44.7i)T^{2} \)
59 \( 1 + (-3.89 - 6.14i)T + (-25.1 + 53.3i)T^{2} \)
61 \( 1 + (10.4 - 4.14i)T + (44.4 - 41.7i)T^{2} \)
67 \( 1 + (7.54 - 8.02i)T + (-4.20 - 66.8i)T^{2} \)
71 \( 1 + (2.21 - 11.5i)T + (-66.0 - 26.1i)T^{2} \)
73 \( 1 + (-6.33 - 4.01i)T + (31.0 + 66.0i)T^{2} \)
79 \( 1 + (3.38 + 7.19i)T + (-50.3 + 60.8i)T^{2} \)
83 \( 1 + (-3.34 - 1.57i)T + (52.9 + 63.9i)T^{2} \)
89 \( 1 + (-0.422 + 0.666i)T + (-37.8 - 80.5i)T^{2} \)
97 \( 1 + (9.86 + 10.5i)T + (-6.09 + 96.8i)T^{2} \)
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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

−10.61103994859723915785272938120, −10.17006717544757224109454856945, −8.894373764443390658126402328398, −8.314703903121974687738302755178, −7.06739333728523889235208652672, −6.48182147798618162222283491096, −5.74303784830381603184409581951, −4.03493886024809245672654775706, −3.33721500620378654340508290214, −1.59670183489245265684688355166, 0.33920865357484616446936231552, 1.70508466849583736537339986374, 3.26554533859599184422796508974, 4.70943675349033241559889797164, 5.57502409603371689289340156396, 6.53686759209064247990682200871, 7.61887207853472108856330940142, 8.223121381177010148361281431726, 9.259702302476369008297035093679, 9.789904887417558699818073758932

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