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Properties

Label 2-750-125.106-c1-0-14
Degree $2$
Conductor $750$
Sign $0.492 + 0.870i$
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 + (−2.16 + 0.555i)5-s + (0.187 + 0.982i)6-s + (2.84 − 2.06i)7-s + (0.992 + 0.125i)8-s + (0.0627 − 0.998i)9-s + (0.691 − 2.12i)10-s + (1.03 − 1.63i)11-s + (−0.929 − 0.368i)12-s + (−0.0703 + 1.11i)13-s + (0.221 + 3.51i)14-s + (−1.19 + 1.88i)15-s + (−0.637 + 0.770i)16-s + (−0.395 + 0.841i)17-s + ⋯
L(s)  = 1  + (−0.378 + 0.597i)2-s + (0.420 − 0.395i)3-s + (−0.212 − 0.452i)4-s + (−0.968 + 0.248i)5-s + (0.0764 + 0.401i)6-s + (1.07 − 0.782i)7-s + (0.350 + 0.0443i)8-s + (0.0209 − 0.332i)9-s + (0.218 − 0.672i)10-s + (0.313 − 0.493i)11-s + (−0.268 − 0.106i)12-s + (−0.0195 + 0.310i)13-s + (0.0590 + 0.939i)14-s + (−0.309 + 0.487i)15-s + (−0.159 + 0.192i)16-s + (−0.0959 + 0.203i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.492 + 0.870i$
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: \((2,\ 750,\ (\ :1/2),\ 0.492 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.973700 - 0.568048i\)
\(L(\frac12)\) \(\approx\) \(0.973700 - 0.568048i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 - 0.555i)T \)
good7 \( 1 + (-2.84 + 2.06i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-1.03 + 1.63i)T + (-4.68 - 9.95i)T^{2} \)
13 \( 1 + (0.0703 - 1.11i)T + (-12.8 - 1.62i)T^{2} \)
17 \( 1 + (0.395 - 0.841i)T + (-10.8 - 13.0i)T^{2} \)
19 \( 1 + (5.26 + 4.94i)T + (1.19 + 18.9i)T^{2} \)
23 \( 1 + (7.88 + 2.02i)T + (20.1 + 11.0i)T^{2} \)
29 \( 1 + (-4.39 - 2.41i)T + (15.5 + 24.4i)T^{2} \)
31 \( 1 + (-2.50 + 5.31i)T + (-19.7 - 23.8i)T^{2} \)
37 \( 1 + (-7.43 + 8.99i)T + (-6.93 - 36.3i)T^{2} \)
41 \( 1 + (-3.03 + 0.778i)T + (35.9 - 19.7i)T^{2} \)
43 \( 1 + (0.106 - 0.327i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-5.06 + 0.640i)T + (45.5 - 11.6i)T^{2} \)
53 \( 1 + (-2.05 + 10.7i)T + (-49.2 - 19.5i)T^{2} \)
59 \( 1 + (1.34 + 0.532i)T + (43.0 + 40.3i)T^{2} \)
61 \( 1 + (-0.450 - 0.115i)T + (53.4 + 29.3i)T^{2} \)
67 \( 1 + (4.44 - 2.44i)T + (35.9 - 56.5i)T^{2} \)
71 \( 1 + (7.84 - 0.991i)T + (68.7 - 17.6i)T^{2} \)
73 \( 1 + (-6.35 + 2.51i)T + (53.2 - 49.9i)T^{2} \)
79 \( 1 + (5.08 - 4.77i)T + (4.96 - 78.8i)T^{2} \)
83 \( 1 + (6.64 + 6.24i)T + (5.21 + 82.8i)T^{2} \)
89 \( 1 + (8.59 - 3.40i)T + (64.8 - 60.9i)T^{2} \)
97 \( 1 + (-15.5 - 8.52i)T + (51.9 + 81.8i)T^{2} \)
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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

−10.26550118835752962554505674605, −8.972907863717227108757846038534, −8.295406484358676785063258585940, −7.73408944234309341014838882845, −6.94614519182714713634011389912, −6.09704273847940222483559108155, −4.50667280437475496920433136536, −4.01469965799012579478805207809, −2.28265760368180012187699106672, −0.65320327233500266220693855849, 1.60740214078989590128268045851, 2.80343127842448666410438740559, 4.14443296225433741379691375538, 4.61574530895142291219414923562, 5.98147528069283508207462540858, 7.50073603373162949993112898022, 8.301372379815247286793030585618, 8.533461422357577231333566333003, 9.701944903573618952678670485246, 10.46590723762465618678724070051

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