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Properties

Label 2-750-125.106-c1-0-15
Degree $2$
Conductor $750$
Sign $-0.450 + 0.892i$
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  + (−0.535 + 0.844i)2-s + (−0.728 + 0.684i)3-s + (−0.425 − 0.904i)4-s + (−0.967 − 2.01i)5-s + (−0.187 − 0.982i)6-s + (1.09 − 0.795i)7-s + (0.992 + 0.125i)8-s + (0.0627 − 0.998i)9-s + (2.22 + 0.263i)10-s + (−1.13 + 1.79i)11-s + (0.929 + 0.368i)12-s + (−0.332 + 5.27i)13-s + (0.0849 + 1.35i)14-s + (2.08 + 0.807i)15-s + (−0.637 + 0.770i)16-s + (2.14 − 4.55i)17-s + ⋯
L(s)  = 1  + (−0.378 + 0.597i)2-s + (−0.420 + 0.395i)3-s + (−0.212 − 0.452i)4-s + (−0.432 − 0.901i)5-s + (−0.0764 − 0.401i)6-s + (0.413 − 0.300i)7-s + (0.350 + 0.0443i)8-s + (0.0209 − 0.332i)9-s + (0.702 + 0.0832i)10-s + (−0.343 + 0.541i)11-s + (0.268 + 0.106i)12-s + (−0.0921 + 1.46i)13-s + (0.0227 + 0.360i)14-s + (0.538 + 0.208i)15-s + (−0.159 + 0.192i)16-s + (0.519 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.450 + 0.892i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.450 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.141681 - 0.230090i\)
\(L(\frac12)\) \(\approx\) \(0.141681 - 0.230090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.535 - 0.844i)T \)
3 \( 1 + (0.728 - 0.684i)T \)
5 \( 1 + (0.967 + 2.01i)T \)
good7 \( 1 + (-1.09 + 0.795i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.13 - 1.79i)T + (-4.68 - 9.95i)T^{2} \)
13 \( 1 + (0.332 - 5.27i)T + (-12.8 - 1.62i)T^{2} \)
17 \( 1 + (-2.14 + 4.55i)T + (-10.8 - 13.0i)T^{2} \)
19 \( 1 + (2.74 + 2.58i)T + (1.19 + 18.9i)T^{2} \)
23 \( 1 + (5.44 + 1.39i)T + (20.1 + 11.0i)T^{2} \)
29 \( 1 + (8.48 + 4.66i)T + (15.5 + 24.4i)T^{2} \)
31 \( 1 + (-2.52 + 5.37i)T + (-19.7 - 23.8i)T^{2} \)
37 \( 1 + (6.41 - 7.75i)T + (-6.93 - 36.3i)T^{2} \)
41 \( 1 + (4.53 - 1.16i)T + (35.9 - 19.7i)T^{2} \)
43 \( 1 + (-2.95 + 9.08i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-6.44 + 0.814i)T + (45.5 - 11.6i)T^{2} \)
53 \( 1 + (-1.68 + 8.84i)T + (-49.2 - 19.5i)T^{2} \)
59 \( 1 + (12.0 + 4.78i)T + (43.0 + 40.3i)T^{2} \)
61 \( 1 + (7.84 + 2.01i)T + (53.4 + 29.3i)T^{2} \)
67 \( 1 + (-0.312 + 0.171i)T + (35.9 - 56.5i)T^{2} \)
71 \( 1 + (-4.80 + 0.606i)T + (68.7 - 17.6i)T^{2} \)
73 \( 1 + (-4.27 + 1.69i)T + (53.2 - 49.9i)T^{2} \)
79 \( 1 + (8.06 - 7.57i)T + (4.96 - 78.8i)T^{2} \)
83 \( 1 + (-2.78 - 2.61i)T + (5.21 + 82.8i)T^{2} \)
89 \( 1 + (8.28 - 3.27i)T + (64.8 - 60.9i)T^{2} \)
97 \( 1 + (-5.49 - 3.02i)T + (51.9 + 81.8i)T^{2} \)
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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

−9.754809389773222257375955471372, −9.329177850817729201684231150305, −8.324585558720250293768951930570, −7.52797904544276252337383835043, −6.67680967055910250779241994004, −5.49807932294221928297244893409, −4.65862901293388403508798327997, −4.05794064177202220218343949716, −1.88620916254035023819126342596, −0.16207602382448621320023848841, 1.70102069003967489102699522692, 2.99972169842013169310682597332, 3.89437658500745424200955198491, 5.45867299590056763555363827621, 6.13809192586342042097659525740, 7.52863062677466532339302381975, 7.924529176285036440526892737163, 8.808246481620429800067840584979, 10.32543827092717374513433575589, 10.53579480743387202763404417635

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