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Properties

Label 2-750-25.16-c1-0-12
Degree $2$
Conductor $750$
Sign $-0.0918 + 0.995i$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s − 4.80·7-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.714 + 0.518i)11-s + (−0.809 + 0.587i)12-s + (2.28 − 1.66i)13-s + (3.88 + 2.82i)14-s + (−0.809 + 0.587i)16-s + (0.512 − 1.57i)17-s + 0.999·18-s + (1.66 − 5.11i)19-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.126 − 0.388i)6-s − 1.81·7-s + (0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s + (0.215 + 0.156i)11-s + (−0.233 + 0.169i)12-s + (0.633 − 0.460i)13-s + (1.03 + 0.755i)14-s + (−0.202 + 0.146i)16-s + (0.124 − 0.382i)17-s + 0.235·18-s + (0.381 − 1.17i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0918 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0918 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.0918 + 0.995i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.0918 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446573 - 0.489670i\)
\(L(\frac12)\) \(\approx\) \(0.446573 - 0.489670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 \)
good7 \( 1 + 4.80T + 7T^{2} \)
11 \( 1 + (-0.714 - 0.518i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.28 + 1.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.512 + 1.57i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.66 + 5.11i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (4.73 + 3.44i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.10 - 3.40i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.22 + 9.93i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.41 + 1.02i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.40 + 1.01i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.27T + 43T^{2} \)
47 \( 1 + (2.69 + 8.29i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.09 - 3.37i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.37 - 6.08i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.0697 - 0.0506i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3.69 + 11.3i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.08 + 3.33i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.84 + 4.24i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.88 + 11.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.10 - 12.6i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (15.1 + 10.9i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.64 - 17.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.00728537890926820837188957146, −9.400880935599746163068212119412, −8.774708356523749232271703080379, −7.64657928054718808475102843237, −6.64009610536891043709084414885, −5.84397212819176627781006614712, −4.33658414133840494974234038374, −3.35948721681122320768662243353, −2.57040980000610492131812701570, −0.41817027801068038206068789096, 1.37546251363167825078517312475, 2.97510673275951447370715201286, 3.93867580062444276989696517281, 5.78894970323570652136246043140, 6.28475824446035237585283314170, 7.03358042818455331217846278846, 8.035447414604795073164971667248, 8.827179430060302675997489824159, 9.741750833019666596885857150952, 10.17070968926134587573612326797

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