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Properties

Label 2-750-25.9-c1-0-4
Degree $2$
Conductor $750$
Sign $0.357 - 0.934i$
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  + (−0.587 + 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.309 + 0.951i)6-s + 1.72i·7-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (1.97 + 1.43i)11-s + (−0.587 − 0.809i)12-s + (1.74 + 2.40i)13-s + (−1.39 − 1.01i)14-s + (−0.809 + 0.587i)16-s + (−3.18 − 1.03i)17-s + 0.999i·18-s + (−0.694 + 2.13i)19-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (0.549 − 0.178i)3-s + (−0.154 − 0.475i)4-s + (−0.126 + 0.388i)6-s + 0.652i·7-s + (0.336 + 0.109i)8-s + (0.269 − 0.195i)9-s + (0.595 + 0.432i)11-s + (−0.169 − 0.233i)12-s + (0.484 + 0.666i)13-s + (−0.373 − 0.271i)14-s + (−0.202 + 0.146i)16-s + (−0.771 − 0.250i)17-s + 0.235i·18-s + (−0.159 + 0.490i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.357 - 0.934i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.357 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23411 + 0.849494i\)
\(L(\frac12)\) \(\approx\) \(1.23411 + 0.849494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 \)
good7 \( 1 - 1.72iT - 7T^{2} \)
11 \( 1 + (-1.97 - 1.43i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.74 - 2.40i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.18 + 1.03i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.694 - 2.13i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-5.35 + 7.37i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-2.89 - 8.91i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.89 - 5.82i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.94 - 2.67i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.95 + 3.60i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 7.06iT - 43T^{2} \)
47 \( 1 + (4.74 - 1.54i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.51 + 2.11i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.147 - 0.107i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.16 + 3.02i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-11.3 - 3.67i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.51 + 10.8i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.95 + 5.44i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.50 - 10.8i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (8.69 + 2.82i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (6.56 + 4.77i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (9.40 - 3.05i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.39813198583155098372578680184, −9.165705190278488292063832615654, −8.943019689987708103627032740082, −8.108634834216222254700121282973, −6.81457442738438463917675922575, −6.58640177543364618867879860374, −5.17810278834073801295312472618, −4.20680463810306795495797066582, −2.76673637492422845375494094886, −1.47113684634187074355885584782, 0.941820500948988380039574975311, 2.43998018335679180472454065008, 3.60484705805463325170358217328, 4.30056813805594841825974679733, 5.72900568405888997317337196051, 6.94435963017684097356654004764, 7.79305303480865188581848191228, 8.635079148046689919479932676674, 9.370918420001046635013908422723, 10.11329034989324126871484555296

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