Properties

Label 2-750-125.104-c1-0-0
Degree $2$
Conductor $750$
Sign $-0.313 - 0.949i$
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  + (−0.248 − 0.968i)2-s + (0.982 − 0.187i)3-s + (−0.876 + 0.481i)4-s + (−1.84 − 1.26i)5-s + (−0.425 − 0.904i)6-s + (−0.846 + 1.16i)7-s + (0.684 + 0.728i)8-s + (0.929 − 0.368i)9-s + (−0.767 + 2.10i)10-s + (−3.75 + 0.965i)11-s + (−0.770 + 0.637i)12-s + (−0.488 − 1.23i)13-s + (1.33 + 0.529i)14-s + (−2.04 − 0.897i)15-s + (0.535 − 0.844i)16-s + (−1.36 + 2.48i)17-s + ⋯
L(s)  = 1  + (−0.175 − 0.684i)2-s + (0.567 − 0.108i)3-s + (−0.438 + 0.240i)4-s + (−0.824 − 0.565i)5-s + (−0.173 − 0.369i)6-s + (−0.319 + 0.440i)7-s + (0.242 + 0.257i)8-s + (0.309 − 0.122i)9-s + (−0.242 + 0.664i)10-s + (−1.13 + 0.291i)11-s + (−0.222 + 0.184i)12-s + (−0.135 − 0.342i)13-s + (0.357 + 0.141i)14-s + (−0.528 − 0.231i)15-s + (0.133 − 0.211i)16-s + (−0.331 + 0.603i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.313 - 0.949i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.313 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107217 + 0.148349i\)
\(L(\frac12)\) \(\approx\) \(0.107217 + 0.148349i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.248 + 0.968i)T \)
3 \( 1 + (-0.982 + 0.187i)T \)
5 \( 1 + (1.84 + 1.26i)T \)
good7 \( 1 + (0.846 - 1.16i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (3.75 - 0.965i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (0.488 + 1.23i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (1.36 - 2.48i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (0.522 - 2.73i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (5.36 + 0.337i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (5.66 + 0.715i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (2.11 + 1.16i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (-7.23 - 4.59i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (-0.225 - 3.58i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (7.53 - 2.44i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-0.752 + 0.801i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (-0.401 - 0.188i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (8.28 + 10.0i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (-0.204 + 3.25i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-0.359 - 2.84i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (-1.31 - 1.23i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (10.3 + 8.56i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (2.41 + 12.6i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (-4.65 - 0.887i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-5.01 + 6.05i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (2.44 - 19.3i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.53121128329049750176386368856, −9.752221561749802590568362406462, −8.935011291413798022526266192202, −7.922593320117905104330274151190, −7.78989289910782770817362983696, −6.14623491008624666911888565181, −4.96243899371294572260923547909, −3.98569482552763735584596412888, −3.00875169840780722963762272518, −1.83463029766052500492916146386, 0.087470217802395289414392197215, 2.48346395692032881877667601799, 3.64207422354863882225124931943, 4.53661047098643663543187414010, 5.73372658402346866421994588451, 6.95966132015075822311884791306, 7.45043990064628702140702464035, 8.219014213583307459411284930807, 9.098023615275677011194675742688, 10.03462675622473529145401475814

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