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Properties

Label 2-750-75.17-c1-0-9
Degree $2$
Conductor $750$
Sign $0.923 - 0.383i$
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  + (−0.891 − 0.453i)2-s + (−1.36 + 1.06i)3-s + (0.587 + 0.809i)4-s + (1.70 − 0.323i)6-s + (3.13 + 3.13i)7-s + (−0.156 − 0.987i)8-s + (0.750 − 2.90i)9-s + (3.57 − 1.16i)11-s + (−1.66 − 0.484i)12-s + (−1.78 − 3.49i)13-s + (−1.37 − 4.21i)14-s + (−0.309 + 0.951i)16-s + (0.406 − 0.0644i)17-s + (−1.98 + 2.24i)18-s + (2.93 − 4.04i)19-s + ⋯
L(s)  = 1  + (−0.630 − 0.321i)2-s + (−0.790 + 0.612i)3-s + (0.293 + 0.404i)4-s + (0.694 − 0.132i)6-s + (1.18 + 1.18i)7-s + (−0.0553 − 0.349i)8-s + (0.250 − 0.968i)9-s + (1.07 − 0.350i)11-s + (−0.480 − 0.139i)12-s + (−0.494 − 0.969i)13-s + (−0.366 − 1.12i)14-s + (−0.0772 + 0.237i)16-s + (0.0986 − 0.0156i)17-s + (−0.468 + 0.529i)18-s + (0.674 − 0.928i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.923 - 0.383i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.923 - 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02604 + 0.204706i\)
\(L(\frac12)\) \(\approx\) \(1.02604 + 0.204706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 + 0.453i)T \)
3 \( 1 + (1.36 - 1.06i)T \)
5 \( 1 \)
good7 \( 1 + (-3.13 - 3.13i)T + 7iT^{2} \)
11 \( 1 + (-3.57 + 1.16i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.78 + 3.49i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.406 + 0.0644i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-2.93 + 4.04i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.28 - 4.48i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (-1.49 + 1.08i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.69 - 1.96i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-4.70 + 2.39i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (3.24 + 1.05i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-3.71 + 3.71i)T - 43iT^{2} \)
47 \( 1 + (-0.808 + 5.10i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-8.20 - 1.29i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (1.73 - 5.34i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.43 - 13.6i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (0.968 + 6.11i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (-0.992 - 1.36i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.19 - 1.62i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-6.24 - 8.60i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.10 - 7.00i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-0.324 - 0.999i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (7.61 + 1.20i)T + (92.2 + 29.9i)T^{2} \)
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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

−10.47061982354750904823139352182, −9.508793086810981152653522306190, −8.925159592354579729581456576794, −8.042825128976649204788238421061, −6.96788240673382127890858135209, −5.75080266818408087826206096157, −5.17681998817705894796087714016, −3.96366080285122150118154874044, −2.62542504866263963502968258670, −1.07678849915719488363085158626, 1.02263493965088956058510686217, 1.91773329836836088211944838761, 4.18394980941407790800998258382, 4.89075059105535020846541354287, 6.20221818607816991190888917183, 6.89946943977039945018928899778, 7.64952194955061352087224349571, 8.286895413271156179998029508689, 9.584633529228480962891627761566, 10.30223465583322970532347575959

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