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Properties

Label 2-750-75.53-c1-0-14
Degree $2$
Conductor $750$
Sign $0.800 - 0.598i$
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  + (−0.891 + 0.453i)2-s + (1.11 + 1.32i)3-s + (0.587 − 0.809i)4-s + (−1.59 − 0.679i)6-s + (2.03 − 2.03i)7-s + (−0.156 + 0.987i)8-s + (−0.529 + 2.95i)9-s + (−2.60 − 0.847i)11-s + (1.72 − 0.118i)12-s + (2.68 − 5.27i)13-s + (−0.891 + 2.74i)14-s + (−0.309 − 0.951i)16-s + (5.91 + 0.936i)17-s + (−0.868 − 2.87i)18-s + (3.04 + 4.19i)19-s + ⋯
L(s)  = 1  + (−0.630 + 0.321i)2-s + (0.641 + 0.767i)3-s + (0.293 − 0.404i)4-s + (−0.650 − 0.277i)6-s + (0.770 − 0.770i)7-s + (−0.0553 + 0.349i)8-s + (−0.176 + 0.984i)9-s + (−0.786 − 0.255i)11-s + (0.498 − 0.0341i)12-s + (0.744 − 1.46i)13-s + (−0.238 + 0.733i)14-s + (−0.0772 − 0.237i)16-s + (1.43 + 0.227i)17-s + (−0.204 − 0.676i)18-s + (0.699 + 0.962i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.800 - 0.598i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.800 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48964 + 0.495433i\)
\(L(\frac12)\) \(\approx\) \(1.48964 + 0.495433i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 - 0.453i)T \)
3 \( 1 + (-1.11 - 1.32i)T \)
5 \( 1 \)
good7 \( 1 + (-2.03 + 2.03i)T - 7iT^{2} \)
11 \( 1 + (2.60 + 0.847i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.68 + 5.27i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-5.91 - 0.936i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (-3.04 - 4.19i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.515 - 1.01i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (2.34 + 1.70i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-4.56 + 3.31i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-7.18 - 3.66i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (5.02 - 1.63i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (3.10 + 3.10i)T + 43iT^{2} \)
47 \( 1 + (0.726 + 4.58i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-5.29 + 0.837i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (-0.912 - 2.80i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.41 - 4.36i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (0.0721 - 0.455i)T + (-63.7 - 20.7i)T^{2} \)
71 \( 1 + (-3.36 + 4.62i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (8.09 - 4.12i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (6.90 - 9.50i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.88 - 11.9i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (0.402 - 1.23i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.52 + 1.03i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.16692025351360244687707576536, −9.875372416380488035688574595016, −8.432482667342609397889728768499, −7.996907795842721508299907900811, −7.52357242357693675491967878913, −5.80257985123980246581464882436, −5.21195028245753158524367852866, −3.87017224711792604679436881385, −2.91211799293416786434382478042, −1.19711838406454950852600117965, 1.28142781271940643219878882348, 2.31152895200954880979154716768, 3.31347763970275720466510344198, 4.80908993386737030688845838891, 6.02776928599620723460376682891, 7.11323800710456878819929753926, 7.80041929426570367133625592319, 8.627902338987932683026079653670, 9.189273324752547880947604233242, 10.08431053716858195125958414564

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