Properties

Label 2-750-25.11-c1-0-4
Degree $2$
Conductor $750$
Sign $0.805 - 0.592i$
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 + 2.70·7-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (−4.54 + 3.30i)11-s + (−0.809 − 0.587i)12-s + (3.91 + 2.84i)13-s + (−2.19 + 1.59i)14-s + (−0.809 − 0.587i)16-s + (0.323 + 0.994i)17-s + 0.999·18-s + (2.59 + 7.97i)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 + 1.02·7-s + (0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + (−1.37 + 0.996i)11-s + (−0.233 − 0.169i)12-s + (1.08 + 0.789i)13-s + (−0.585 + 0.425i)14-s + (−0.202 − 0.146i)16-s + (0.0783 + 0.241i)17-s + 0.235·18-s + (0.594 + 1.82i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24414 + 0.408037i\)
\(L(\frac12)\) \(\approx\) \(1.24414 + 0.408037i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.70T + 7T^{2} \)
11 \( 1 + (4.54 - 3.30i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.91 - 2.84i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.323 - 0.994i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.59 - 7.97i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-4.28 + 3.11i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.29 + 3.98i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.72 + 5.30i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.42 - 1.75i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.27 - 0.927i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.29T + 43T^{2} \)
47 \( 1 + (0.949 - 2.92i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.90 + 5.87i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-11.4 - 8.33i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.218 - 0.159i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.94 - 6.00i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.40 - 4.31i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.60 + 4.79i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.85 + 11.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.0415 + 0.127i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (9.53 - 6.93i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.815 - 2.51i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.36516892941212016646746732136, −9.528211857868232728008411080138, −8.373467609090364285691269819731, −7.973575670136541701731159668307, −7.22032498210546051850863773408, −6.13290508736531991398295427599, −5.25542814099191686310335651895, −4.11096783255142614562606988545, −2.37280485074117234746475233071, −1.36162412449107342300609611329, 0.921411858983269402580375462134, 2.67005433781728781890117692045, 3.43808676188391083283692866043, 4.95064547303919728788383499263, 5.49374286170138324845397446600, 7.06670480291286970253805156556, 8.027938681139898383322900834850, 8.581802317336256681571001904842, 9.328459711769699622886859492496, 10.49537964333023968603476210321

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