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Properties

Label 2-750-125.104-c1-0-25
Degree $2$
Conductor $750$
Sign $-0.895 - 0.445i$
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 + (−0.411 − 2.19i)5-s + (0.425 + 0.904i)6-s + (0.616 − 0.848i)7-s + (0.684 + 0.728i)8-s + (0.929 − 0.368i)9-s + (−2.02 + 0.945i)10-s + (−0.677 + 0.173i)11-s + (0.770 − 0.637i)12-s + (−1.44 − 3.63i)13-s + (−0.975 − 0.386i)14-s + (0.816 + 2.08i)15-s + (0.535 − 0.844i)16-s + (−2.73 + 4.98i)17-s + ⋯
L(s)  = 1  + (−0.175 − 0.684i)2-s + (−0.567 + 0.108i)3-s + (−0.438 + 0.240i)4-s + (−0.184 − 0.982i)5-s + (0.173 + 0.369i)6-s + (0.233 − 0.320i)7-s + (0.242 + 0.257i)8-s + (0.309 − 0.122i)9-s + (−0.640 + 0.299i)10-s + (−0.204 + 0.0524i)11-s + (0.222 − 0.184i)12-s + (−0.399 − 1.00i)13-s + (−0.260 − 0.103i)14-s + (0.210 + 0.537i)15-s + (0.133 − 0.211i)16-s + (−0.664 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.895 - 0.445i$
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: \((2,\ 750,\ (\ :1/2),\ -0.895 - 0.445i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.104366 + 0.443785i\)
\(L(\frac12)\) \(\approx\) \(0.104366 + 0.443785i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.411 + 2.19i)T \)
good7 \( 1 + (-0.616 + 0.848i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.677 - 0.173i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (1.44 + 3.63i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (2.73 - 4.98i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (-1.10 + 5.80i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-1.10 - 0.0694i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (-6.02 - 0.761i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (8.05 + 4.43i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (6.02 + 3.82i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (-0.639 - 10.1i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (7.95 - 2.58i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (5.22 - 5.56i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (11.4 + 5.36i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (-1.67 - 2.02i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (0.107 - 1.71i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (0.832 + 6.58i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (11.3 + 10.6i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (-5.63 - 4.66i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (1.20 + 6.32i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (-3.39 - 0.646i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-7.02 + 8.49i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (-1.13 + 8.96i)T + (-93.9 - 24.1i)T^{2} \)
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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

−9.964519200038836543526716518325, −9.128704747059674192380842988934, −8.271924491432379167214031018280, −7.47984990489825226450001804647, −6.17105826004781134739884195874, −5.00392063218485884176164844891, −4.52500406468251874548666700974, −3.22769958157932709664144076594, −1.62992594199242172420130728769, −0.26433293256928319571522531942, 1.98653097137071716598378697723, 3.50820155862198738176803396605, 4.78863758328654641597111078886, 5.60334121412111797268764918618, 6.79326863786688822962307483267, 7.02286702832330984435675624512, 8.122398417550392891186265532519, 9.092761384652141650472333898272, 10.04413412966159471931445224540, 10.73534886848765420816530511919

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