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Properties

Label 2-750-25.4-c1-0-0
Degree $2$
Conductor $750$
Sign $-0.720 - 0.693i$
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  + (−0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s + 2i·7-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (−1.61 + 4.97i)11-s + (0.951 − 0.309i)12-s + (−4.61 + 1.5i)13-s + (0.618 − 1.90i)14-s + (0.309 + 0.951i)16-s + (−4.61 − 6.35i)17-s + i·18-s + (−2.23 + 1.62i)19-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.339 − 0.467i)3-s + (0.404 + 0.293i)4-s + (−0.330 + 0.239i)6-s + 0.755i·7-s + (−0.207 − 0.286i)8-s + (−0.103 − 0.317i)9-s + (−0.487 + 1.50i)11-s + (0.274 − 0.0892i)12-s + (−1.28 + 0.416i)13-s + (0.165 − 0.508i)14-s + (0.0772 + 0.237i)16-s + (−1.11 − 1.54i)17-s + 0.235i·18-s + (−0.512 + 0.372i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.720 - 0.693i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.720 - 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102275 + 0.253891i\)
\(L(\frac12)\) \(\approx\) \(0.102275 + 0.253891i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + (1.61 - 4.97i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (4.61 - 1.5i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.61 + 6.35i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.23 - 1.62i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (5.70 + 1.85i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (1.11 + 0.812i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3 - 2.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.04 + 0.663i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.88 + 5.79i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.23iT - 43T^{2} \)
47 \( 1 + (-2.80 + 3.85i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (5.03 - 6.92i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.76 + 8.50i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.73 - 8.42i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-5.70 - 7.85i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-11.4 - 8.33i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.99 - 0.972i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.52 - 4.85i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.427 - 1.31i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (8.14 - 11.2i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.48650001052324870757033048918, −9.594250361422374467924424067232, −9.160936474790617357241803151047, −8.102652895878175641796690891990, −7.27697009152616590932262032213, −6.71460544956799618770765583920, −5.30152711871963398465417370029, −4.26801703449707484684274055689, −2.42780217796635724100898570996, −2.18269711460888378422260565610, 0.14887858938032036473258953315, 2.11542063169241174509695943001, 3.40379260333473074112982901863, 4.48659376697171135755561305992, 5.70283003883363695596698615609, 6.57814563580427121619202893197, 7.79537432033642092099820755275, 8.219021331833674632555401284725, 9.170585788272949369981440519713, 10.03587398533084649180689920805

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