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Properties

Label 2-750-125.104-c1-0-1
Degree $2$
Conductor $750$
Sign $-0.642 + 0.766i$
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 + (1.31 + 1.80i)5-s + (−0.425 − 0.904i)6-s + (−2.62 + 3.61i)7-s + (−0.684 − 0.728i)8-s + (0.929 − 0.368i)9-s + (−1.42 + 1.72i)10-s + (−2.57 + 0.661i)11-s + (0.770 − 0.637i)12-s + (−2.40 − 6.07i)13-s + (−4.15 − 1.64i)14-s + (−1.63 − 1.52i)15-s + (0.535 − 0.844i)16-s + (−0.955 + 1.73i)17-s + ⋯
L(s)  = 1  + (0.175 + 0.684i)2-s + (−0.567 + 0.108i)3-s + (−0.438 + 0.240i)4-s + (0.589 + 0.807i)5-s + (−0.173 − 0.369i)6-s + (−0.993 + 1.36i)7-s + (−0.242 − 0.257i)8-s + (0.309 − 0.122i)9-s + (−0.449 + 0.545i)10-s + (−0.777 + 0.199i)11-s + (0.222 − 0.184i)12-s + (−0.667 − 1.68i)13-s + (−1.11 − 0.439i)14-s + (−0.421 − 0.394i)15-s + (0.133 − 0.211i)16-s + (−0.231 + 0.421i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.642 + 0.766i$
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: \((2,\ 750,\ (\ :1/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.208371 - 0.446860i\)
\(L(\frac12)\) \(\approx\) \(0.208371 - 0.446860i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 - 1.80i)T \)
good7 \( 1 + (2.62 - 3.61i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (2.57 - 0.661i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (2.40 + 6.07i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (0.955 - 1.73i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (0.0553 - 0.290i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-3.01 - 0.189i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (-2.89 - 0.365i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (-5.11 - 2.81i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (9.13 + 5.79i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (0.692 + 11.0i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (4.82 - 1.56i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (4.77 - 5.08i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (7.18 + 3.38i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (4.20 + 5.07i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (0.943 - 14.9i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-1.22 - 9.70i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (1.52 + 1.43i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (-8.66 - 7.16i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (-1.02 - 5.37i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (12.8 + 2.45i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-1.06 + 1.28i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (0.574 - 4.54i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.53410313978160053820597980174, −10.17378318657104579469836590231, −9.232788121881159509725526899785, −8.268758295009431000806873415298, −7.17050417587413537656145792313, −6.40399405239566257077587094493, −5.56087479741301072464236940483, −5.16813658976998594823940563509, −3.30458488786852878244055145902, −2.53362667009020534260011215949, 0.24526139450386339660098327936, 1.63818691810487984914175845933, 3.11667238788513545123665914810, 4.51678324528970514704305405957, 4.89777051547547994924354115122, 6.35607261907568259937214217758, 6.90052908860083691583322962015, 8.186579519998430077886144316091, 9.420154356981351073458807001271, 9.835617926066900828024471731068

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