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Properties

Label 2-750-125.104-c1-0-16
Degree $2$
Conductor $750$
Sign $0.165 + 0.986i$
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 + (2.06 − 0.860i)5-s + (0.425 + 0.904i)6-s + (1.97 − 2.72i)7-s + (0.684 + 0.728i)8-s + (0.929 − 0.368i)9-s + (−1.34 − 1.78i)10-s + (1.55 − 0.399i)11-s + (0.770 − 0.637i)12-s + (1.38 + 3.49i)13-s + (−3.13 − 1.23i)14-s + (−1.86 + 1.23i)15-s + (0.535 − 0.844i)16-s + (−0.651 + 1.18i)17-s + ⋯
L(s)  = 1  + (−0.175 − 0.684i)2-s + (−0.567 + 0.108i)3-s + (−0.438 + 0.240i)4-s + (0.923 − 0.384i)5-s + (0.173 + 0.369i)6-s + (0.748 − 1.02i)7-s + (0.242 + 0.257i)8-s + (0.309 − 0.122i)9-s + (−0.425 − 0.564i)10-s + (0.469 − 0.120i)11-s + (0.222 − 0.184i)12-s + (0.384 + 0.970i)13-s + (−0.836 − 0.331i)14-s + (−0.481 + 0.318i)15-s + (0.133 − 0.211i)16-s + (−0.158 + 0.287i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.165 + 0.986i$
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: \((2,\ 750,\ (\ :1/2),\ 0.165 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12361 - 0.950457i\)
\(L(\frac12)\) \(\approx\) \(1.12361 - 0.950457i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.06 + 0.860i)T \)
good7 \( 1 + (-1.97 + 2.72i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (-1.55 + 0.399i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (-1.38 - 3.49i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (0.651 - 1.18i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (-0.0105 + 0.0551i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-4.41 - 0.277i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (7.10 + 0.897i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (-0.591 - 0.325i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (1.14 + 0.727i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (-0.00918 - 0.145i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (-11.7 + 3.80i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-8.03 + 8.55i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (4.77 + 2.24i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (9.02 + 10.9i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (-0.242 + 3.86i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-1.60 - 12.6i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (5.49 + 5.15i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (0.766 + 0.633i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (-0.467 - 2.45i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (7.91 + 1.51i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-2.53 + 3.06i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (0.668 - 5.29i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.36134087093055971765493604644, −9.346439945370980399006215453846, −8.834115151097138636881071914261, −7.56166937664941862402382570180, −6.62585951654920206968044972235, −5.55259556831517757112906330623, −4.57532759310530023071948387045, −3.80710575897450406333086513354, −1.99262928891384635255531420531, −1.04109806803584410131642100148, 1.40770601912492310772027899537, 2.80119460077699342662359816495, 4.54013022832365246612598484446, 5.65279795591903508394143655077, 5.84158364780890740030119374442, 6.99772759355996564151273480974, 7.84406077760835381351869431281, 8.983996862074175475521742546398, 9.421809654168891505638949896389, 10.66529336037692334832389622992

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