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Properties

Label 2-750-75.8-c1-0-9
Degree $2$
Conductor $750$
Sign $0.159 - 0.987i$
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 + (1.06 + 1.36i)3-s + (−0.587 − 0.809i)4-s + (1.70 − 0.323i)6-s + (−3.13 + 3.13i)7-s + (−0.987 + 0.156i)8-s + (−0.750 + 2.90i)9-s + (3.57 − 1.16i)11-s + (0.484 − 1.66i)12-s + (−3.49 + 1.78i)13-s + (1.37 + 4.21i)14-s + (−0.309 + 0.951i)16-s + (0.0644 + 0.406i)17-s + (2.24 + 1.98i)18-s + (−2.93 + 4.04i)19-s + ⋯
L(s)  = 1  + (0.321 − 0.630i)2-s + (0.612 + 0.790i)3-s + (−0.293 − 0.404i)4-s + (0.694 − 0.132i)6-s + (−1.18 + 1.18i)7-s + (−0.349 + 0.0553i)8-s + (−0.250 + 0.968i)9-s + (1.07 − 0.350i)11-s + (0.139 − 0.480i)12-s + (−0.969 + 0.494i)13-s + (0.366 + 1.12i)14-s + (−0.0772 + 0.237i)16-s + (0.0156 + 0.0986i)17-s + (0.529 + 0.468i)18-s + (−0.674 + 0.928i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.159 - 0.987i$
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: \((2,\ 750,\ (\ :1/2),\ 0.159 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18378 + 1.00834i\)
\(L(\frac12)\) \(\approx\) \(1.18378 + 1.00834i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.06 - 1.36i)T \)
5 \( 1 \)
good7 \( 1 + (3.13 - 3.13i)T - 7iT^{2} \)
11 \( 1 + (-3.57 + 1.16i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (3.49 - 1.78i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.0644 - 0.406i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (2.93 - 4.04i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-4.48 - 2.28i)T + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (1.49 - 1.08i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.69 - 1.96i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.39 - 4.70i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (3.24 + 1.05i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (3.71 + 3.71i)T + 43iT^{2} \)
47 \( 1 + (-5.10 - 0.808i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (-1.29 + 8.20i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-1.73 + 5.34i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.43 - 13.6i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-6.11 + 0.968i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (-0.992 - 1.36i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.62 + 3.19i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (6.24 + 8.60i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.00 + 1.10i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (0.324 + 0.999i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.20 + 7.61i)T + (-92.2 - 29.9i)T^{2} \)
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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

−10.31285757750890889805751528076, −9.696215451692925383714567213718, −9.060801490660422604677688746407, −8.461374110721475933311649454686, −6.90733845184856142062499305215, −5.93583059739564674072619287054, −4.98121127775139436206759357444, −3.83737925189970677607828892588, −3.08865231297431445895657265531, −2.05982796533536565772523336900, 0.65083481291879481621928985750, 2.61314846911989349175300178776, 3.64931466052737937287498569691, 4.56900572224194806843762980894, 6.09764086945563727623425877109, 6.99527335233489440175554815763, 7.11132110597621503696159456196, 8.271658695644479224826159100795, 9.302650652687483949197142863121, 9.803298584971783075690754201816

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