Properties

Label 2-750-125.89-c1-0-12
Degree $2$
Conductor $750$
Sign $0.611 + 0.791i$
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  + (0.481 − 0.876i)2-s + (−0.368 + 0.929i)3-s + (−0.535 − 0.844i)4-s + (−2.15 − 0.597i)5-s + (0.637 + 0.770i)6-s + (2.46 + 0.800i)7-s + (−0.998 + 0.0627i)8-s + (−0.728 − 0.684i)9-s + (−1.56 + 1.60i)10-s + (1.05 + 0.578i)11-s + (0.982 − 0.187i)12-s + (1.11 − 1.18i)13-s + (1.88 − 1.77i)14-s + (1.34 − 1.78i)15-s + (−0.425 + 0.904i)16-s + (1.15 + 0.735i)17-s + ⋯
L(s)  = 1  + (0.340 − 0.619i)2-s + (−0.212 + 0.536i)3-s + (−0.267 − 0.422i)4-s + (−0.963 − 0.267i)5-s + (0.260 + 0.314i)6-s + (0.931 + 0.302i)7-s + (−0.352 + 0.0221i)8-s + (−0.242 − 0.228i)9-s + (−0.493 + 0.505i)10-s + (0.317 + 0.174i)11-s + (0.283 − 0.0540i)12-s + (0.307 − 0.327i)13-s + (0.504 − 0.473i)14-s + (0.348 − 0.460i)15-s + (−0.106 + 0.226i)16-s + (0.281 + 0.178i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.611 + 0.791i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.611 + 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40016 - 0.687353i\)
\(L(\frac12)\) \(\approx\) \(1.40016 - 0.687353i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.481 + 0.876i)T \)
3 \( 1 + (0.368 - 0.929i)T \)
5 \( 1 + (2.15 + 0.597i)T \)
good7 \( 1 + (-2.46 - 0.800i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.05 - 0.578i)T + (5.89 + 9.28i)T^{2} \)
13 \( 1 + (-1.11 + 1.18i)T + (-0.816 - 12.9i)T^{2} \)
17 \( 1 + (-1.15 - 0.735i)T + (7.23 + 15.3i)T^{2} \)
19 \( 1 + (-4.49 + 1.77i)T + (13.8 - 13.0i)T^{2} \)
23 \( 1 + (0.923 + 7.31i)T + (-22.2 + 5.71i)T^{2} \)
29 \( 1 + (-4.31 + 1.10i)T + (25.4 - 13.9i)T^{2} \)
31 \( 1 + (-1.57 + 2.48i)T + (-13.1 - 28.0i)T^{2} \)
37 \( 1 + (-5.23 - 2.46i)T + (23.5 + 28.5i)T^{2} \)
41 \( 1 + (-1.30 - 0.164i)T + (39.7 + 10.1i)T^{2} \)
43 \( 1 + (-2.63 + 3.62i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (2.31 + 0.145i)T + (46.6 + 5.89i)T^{2} \)
53 \( 1 + (1.84 + 1.52i)T + (9.93 + 52.0i)T^{2} \)
59 \( 1 + (0.630 + 3.30i)T + (-54.8 + 21.7i)T^{2} \)
61 \( 1 + (-10.2 + 1.29i)T + (59.0 - 15.1i)T^{2} \)
67 \( 1 + (1.02 - 3.97i)T + (-58.7 - 32.2i)T^{2} \)
71 \( 1 + (0.246 - 3.92i)T + (-70.4 - 8.89i)T^{2} \)
73 \( 1 + (5.24 + 1.00i)T + (67.8 + 26.8i)T^{2} \)
79 \( 1 + (1.12 + 0.444i)T + (57.5 + 54.0i)T^{2} \)
83 \( 1 + (-6.06 - 15.3i)T + (-60.5 + 56.8i)T^{2} \)
89 \( 1 + (-1.07 + 5.63i)T + (-82.7 - 32.7i)T^{2} \)
97 \( 1 + (-2.10 - 8.20i)T + (-85.0 + 46.7i)T^{2} \)
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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

−10.44338763642619651292134035752, −9.496149372041654564023014972087, −8.515066206939908564992059354072, −7.932985284894569226137774565513, −6.60334477876013098539308953780, −5.36864199507425986727563672031, −4.63981093825975338483764255305, −3.87570890887085245371579736532, −2.67848390756857895000904155159, −0.940009117307165196281610006261, 1.24426715110766537634687311004, 3.14131082174047084998792292358, 4.16339533290245730782249701346, 5.12452158441911659838782672936, 6.13596491929716444765705010262, 7.17623454890520333713792670377, 7.71601168039738646808926523911, 8.352035371155528535020340323140, 9.456920170508105392118815542579, 10.74073104223305080604140979868

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