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Properties

Label 2-750-25.21-c1-0-0
Degree $2$
Conductor $750$
Sign $-0.155 - 0.987i$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)6-s − 3.52·7-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (1.62 − 4.99i)11-s + (0.309 + 0.951i)12-s + (0.191 + 0.588i)13-s + (−1.08 + 3.34i)14-s + (0.309 + 0.951i)16-s + (−2.78 + 2.02i)17-s + 18-s + (−1.83 + 1.33i)19-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.330 + 0.239i)6-s − 1.33·7-s + (−0.286 + 0.207i)8-s + (0.103 + 0.317i)9-s + (0.489 − 1.50i)11-s + (0.0892 + 0.274i)12-s + (0.0530 + 0.163i)13-s + (−0.290 + 0.895i)14-s + (0.0772 + 0.237i)16-s + (−0.676 + 0.491i)17-s + 0.235·18-s + (−0.422 + 0.306i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.155 - 0.987i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.155 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0385151 + 0.0450342i\)
\(L(\frac12)\) \(\approx\) \(0.0385151 + 0.0450342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 \)
good7 \( 1 + 3.52T + 7T^{2} \)
11 \( 1 + (-1.62 + 4.99i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.191 - 0.588i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.78 - 2.02i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.83 - 1.33i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.76 - 8.51i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.16 - 1.57i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (7.90 - 5.74i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.309 + 0.952i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.94 - 5.98i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.51T + 43T^{2} \)
47 \( 1 + (8.63 + 6.27i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.447 + 0.325i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.0861 - 0.265i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.13 + 3.50i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-9.63 + 7.00i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (3.84 + 2.79i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.03 + 9.35i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.27 + 3.83i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.15 - 1.56i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (4.13 - 12.7i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.53 + 2.56i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.80614352994646801190809000500, −9.809156901492626658086931552030, −9.080803462617884935895534856824, −8.166969585237045188939774547815, −6.78275786178858516995817618603, −6.16526012824860632114231594133, −5.34200827284238878256504426345, −3.81923667332080905871365352592, −3.21208178115443152719952831557, −1.60709704905553212143501963535, 0.02912098851634341519522236468, 2.49608180226864730866369915641, 3.96780216956266778479150344870, 4.60160905394438212600674544054, 5.80846666602256244812716247912, 6.68626463385443704669778342889, 7.08002482555958685862200172737, 8.434835783028252275127060247861, 9.458896396151708514361803705443, 9.847667575877045690786974356243

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