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Properties

Label 2-750-25.16-c1-0-0
Degree $2$
Conductor $750$
Sign $-0.570 - 0.821i$
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 − 0.533·7-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (1.16 + 0.843i)11-s + (−0.809 + 0.587i)12-s + (−5.31 + 3.86i)13-s + (0.431 + 0.313i)14-s + (−0.809 + 0.587i)16-s + (−0.296 + 0.911i)17-s + 0.999·18-s + (0.0657 − 0.202i)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 − 0.201·7-s + (0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s + (0.349 + 0.254i)11-s + (−0.233 + 0.169i)12-s + (−1.47 + 1.07i)13-s + (0.115 + 0.0838i)14-s + (−0.202 + 0.146i)16-s + (−0.0718 + 0.221i)17-s + 0.235·18-s + (0.0150 − 0.0464i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.570 - 0.821i$
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: \((2,\ 750,\ (\ :1/2),\ -0.570 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.293384 + 0.560753i\)
\(L(\frac12)\) \(\approx\) \(0.293384 + 0.560753i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.533T + 7T^{2} \)
11 \( 1 + (-1.16 - 0.843i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (5.31 - 3.86i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.296 - 0.911i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.0657 + 0.202i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.04 + 2.21i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.91 - 5.89i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.722 - 2.22i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.28 - 2.38i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.42 + 4.66i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + (-3.13 - 9.65i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.999 + 3.07i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.08 - 4.42i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (10.1 + 7.38i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.13 - 6.57i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-3.12 - 9.62i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-11.3 - 8.21i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.79 - 14.7i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-5.04 + 15.5i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (4.54 + 3.30i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.71 + 5.29i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.46456004977003452114904322847, −9.734420148169723682273640975774, −9.181305774586469419320920455383, −8.306002227429257459842430586181, −7.25793629433346914218782839284, −6.48903855927479343726671601792, −5.00927638649845832435123111179, −4.18225787497095889547314896397, −2.99091338982163912513379938543, −1.84019909874430694966457793108, 0.36055882820692544857914326288, 2.05071022819970515269916054298, 3.24489066548411178494066433294, 4.80816506856845471799760362058, 5.83401538069267216042036760018, 6.67338774845096049027472038950, 7.67178272884312563741506157681, 8.060683193690852213347791259845, 9.257565756524295949396240218505, 9.843385436561807446211958792064

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