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Properties

Label 2-750-75.8-c1-0-20
Degree $2$
Conductor $750$
Sign $-0.173 + 0.984i$
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  + (−0.453 + 0.891i)2-s + (−1.72 + 0.102i)3-s + (−0.587 − 0.809i)4-s + (0.693 − 1.58i)6-s + (2.97 − 2.97i)7-s + (0.987 − 0.156i)8-s + (2.97 − 0.353i)9-s + (−4.73 + 1.53i)11-s + (1.09 + 1.33i)12-s + (−1.57 + 0.801i)13-s + (1.30 + 4.00i)14-s + (−0.309 + 0.951i)16-s + (−0.223 − 1.40i)17-s + (−1.03 + 2.81i)18-s + (1.09 − 1.50i)19-s + ⋯
L(s)  = 1  + (−0.321 + 0.630i)2-s + (−0.998 + 0.0589i)3-s + (−0.293 − 0.404i)4-s + (0.283 − 0.647i)6-s + (1.12 − 1.12i)7-s + (0.349 − 0.0553i)8-s + (0.993 − 0.117i)9-s + (−1.42 + 0.464i)11-s + (0.317 + 0.386i)12-s + (−0.436 + 0.222i)13-s + (0.347 + 1.07i)14-s + (−0.0772 + 0.237i)16-s + (−0.0541 − 0.341i)17-s + (−0.244 + 0.663i)18-s + (0.250 − 0.345i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.276381 - 0.329244i\)
\(L(\frac12)\) \(\approx\) \(0.276381 - 0.329244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.453 - 0.891i)T \)
3 \( 1 + (1.72 - 0.102i)T \)
5 \( 1 \)
good7 \( 1 + (-2.97 + 2.97i)T - 7iT^{2} \)
11 \( 1 + (4.73 - 1.53i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.57 - 0.801i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.223 + 1.40i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (-1.09 + 1.50i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.86 + 0.951i)T + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (4.57 - 3.32i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (5.23 + 3.80i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.16 + 2.29i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (-3.58 - 1.16i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (6.26 + 6.26i)T + 43iT^{2} \)
47 \( 1 + (-4.62 - 0.732i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (-0.801 + 5.05i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-3.44 + 10.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.99 + 9.23i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-6.39 + 1.01i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (8.85 + 12.1i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.19 - 8.23i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (-2.20 - 3.03i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (16.6 - 2.63i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (0.351 + 1.08i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.549 - 3.47i)T + (-92.2 - 29.9i)T^{2} \)
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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

−10.24558941257336772661174383130, −9.408896892425152266528103314547, −8.031613284986930694956217789226, −7.43916069059312547279829318089, −6.85735925098089058942473086104, −5.43584061872440622038961877258, −4.99244362128761684107237911507, −4.05038360904727069203536491572, −1.87327174584292528144055533262, −0.27678371401726546450760423126, 1.60521248514546390234749018311, 2.72278894327301265930719916251, 4.33036829618142678856901928282, 5.40371711643795906055937018401, 5.72310894334969801611405813673, 7.37469607383611409221732849359, 8.040739789910481087755861215765, 8.912089765732135702906900959693, 10.04340831997161729124762171288, 10.67373987412032134441236485292

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