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Properties

Label 2-750-125.104-c1-0-6
Degree $2$
Conductor $750$
Sign $0.911 + 0.411i$
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 + (−2.11 − 0.715i)5-s + (0.425 + 0.904i)6-s + (−0.841 + 1.15i)7-s + (0.684 + 0.728i)8-s + (0.929 − 0.368i)9-s + (−0.166 + 2.22i)10-s + (−2.36 + 0.607i)11-s + (0.770 − 0.637i)12-s + (−0.543 − 1.37i)13-s + (1.33 + 0.527i)14-s + (2.21 + 0.305i)15-s + (0.535 − 0.844i)16-s + (1.63 − 2.97i)17-s + ⋯
L(s)  = 1  + (−0.175 − 0.684i)2-s + (−0.567 + 0.108i)3-s + (−0.438 + 0.240i)4-s + (−0.947 − 0.320i)5-s + (0.173 + 0.369i)6-s + (−0.318 + 0.437i)7-s + (0.242 + 0.257i)8-s + (0.309 − 0.122i)9-s + (−0.0525 + 0.705i)10-s + (−0.713 + 0.183i)11-s + (0.222 − 0.184i)12-s + (−0.150 − 0.380i)13-s + (0.355 + 0.140i)14-s + (0.571 + 0.0790i)15-s + (0.133 − 0.211i)16-s + (0.397 − 0.722i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.911 + 0.411i$
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: \((2,\ 750,\ (\ :1/2),\ 0.911 + 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.735642 - 0.158481i\)
\(L(\frac12)\) \(\approx\) \(0.735642 - 0.158481i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.11 + 0.715i)T \)
good7 \( 1 + (0.841 - 1.15i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (2.36 - 0.607i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (0.543 + 1.37i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (-1.63 + 2.97i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (0.852 - 4.46i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-8.17 - 0.514i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (0.0523 + 0.00661i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (-4.52 - 2.48i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (-2.70 - 1.71i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (0.552 + 8.78i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (-2.16 + 0.704i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-5.05 + 5.38i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (-1.52 - 0.717i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (-1.32 - 1.60i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (-0.190 + 3.02i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (0.799 + 6.32i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (-0.276 - 0.259i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (-8.85 - 7.32i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (-2.80 - 14.6i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (-3.61 - 0.689i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (4.55 - 5.50i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (-1.52 + 12.0i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.46806291067934528786823305032, −9.564157136736879309647202722618, −8.685380079562185264173609510398, −7.80679655080813510776221133993, −6.96760891096724430198491168575, −5.53080693423258927661678923824, −4.86244630910528666568002911144, −3.70799936124221565491511900669, −2.68469296314101332340435912396, −0.830096520262914669442149407890, 0.69037713454060228924746637407, 2.93352869488783056769597608405, 4.22589731778488883896974041939, 5.01863689926246880615559771526, 6.22154416316033663077190341638, 6.97579544808231096559542076398, 7.64237748294431512415960283323, 8.494817437564828188632312964687, 9.517017338053025114219789081882, 10.57625046666507168367938591437

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