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Properties

Label 2-750-125.104-c1-0-17
Degree $2$
Conductor $750$
Sign $0.935 - 0.353i$
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 + (2.23 + 0.142i)5-s + (−0.425 − 0.904i)6-s + (1.72 − 2.37i)7-s + (−0.684 − 0.728i)8-s + (0.929 − 0.368i)9-s + (0.416 + 2.19i)10-s + (−0.703 + 0.180i)11-s + (0.770 − 0.637i)12-s + (0.259 + 0.655i)13-s + (2.72 + 1.08i)14-s + (−2.21 + 0.277i)15-s + (0.535 − 0.844i)16-s + (1.73 − 3.15i)17-s + ⋯
L(s)  = 1  + (0.175 + 0.684i)2-s + (−0.567 + 0.108i)3-s + (−0.438 + 0.240i)4-s + (0.997 + 0.0638i)5-s + (−0.173 − 0.369i)6-s + (0.652 − 0.897i)7-s + (−0.242 − 0.257i)8-s + (0.309 − 0.122i)9-s + (0.131 + 0.694i)10-s + (−0.212 + 0.0544i)11-s + (0.222 − 0.184i)12-s + (0.0719 + 0.181i)13-s + (0.729 + 0.288i)14-s + (−0.572 + 0.0717i)15-s + (0.133 − 0.211i)16-s + (0.421 − 0.765i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.935 - 0.353i$
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: \((2,\ 750,\ (\ :1/2),\ 0.935 - 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65969 + 0.303036i\)
\(L(\frac12)\) \(\approx\) \(1.65969 + 0.303036i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.142i)T \)
good7 \( 1 + (-1.72 + 2.37i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.703 - 0.180i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (-0.259 - 0.655i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (-1.73 + 3.15i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (-1.47 + 7.75i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (0.916 + 0.0576i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (-2.36 - 0.298i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (-1.78 - 0.982i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (0.281 + 0.178i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (-0.271 - 4.32i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (-2.31 + 0.752i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (4.40 - 4.68i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (-5.22 - 2.45i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (-5.95 - 7.19i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (0.964 - 15.3i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (1.52 + 12.0i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (-5.24 - 4.92i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (6.86 + 5.68i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (0.700 + 3.67i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (4.75 + 0.907i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-4.60 + 5.57i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (-0.435 + 3.44i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.36592642779953821603470077574, −9.565489563367589582259678175674, −8.723499240949610032015148277975, −7.47982983566757149226976245298, −6.92750707490651944300763259875, −5.95946696889572118069262427150, −5.02170241359257656194755548969, −4.44292928402846162256357595471, −2.81221939866393411201256122828, −1.03934929684864739383922756928, 1.42577019530941129795651381526, 2.32391379695623072432912653891, 3.75915624299642800174813174656, 5.15875295546831121818079713088, 5.61390905627709342153756773790, 6.42953987243150398810653691475, 7.965260698212146401946631181269, 8.652340742915348432184406471027, 9.822349667900385128432329276300, 10.25247981341024418070621688999

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