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Properties

Label 2-750-75.17-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: $\chi_{750} (107, \cdot )$
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

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

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