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Properties

Label 2-750-125.94-c1-0-2
Degree $2$
Conductor $750$
Sign $-0.0195 - 0.999i$
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  + (0.998 − 0.0627i)2-s + (−0.904 + 0.425i)3-s + (0.992 − 0.125i)4-s + (−2.21 − 0.275i)5-s + (−0.876 + 0.481i)6-s + (−0.591 − 0.813i)7-s + (0.982 − 0.187i)8-s + (0.637 − 0.770i)9-s + (−2.23 − 0.135i)10-s + (0.286 + 4.54i)11-s + (−0.844 + 0.535i)12-s + (0.824 + 0.682i)13-s + (−0.641 − 0.774i)14-s + (2.12 − 0.695i)15-s + (0.968 − 0.248i)16-s + (−0.674 + 5.33i)17-s + ⋯
L(s)  = 1  + (0.705 − 0.0443i)2-s + (−0.522 + 0.245i)3-s + (0.496 − 0.0626i)4-s + (−0.992 − 0.123i)5-s + (−0.357 + 0.196i)6-s + (−0.223 − 0.307i)7-s + (0.347 − 0.0662i)8-s + (0.212 − 0.256i)9-s + (−0.705 − 0.0427i)10-s + (0.0862 + 1.37i)11-s + (−0.243 + 0.154i)12-s + (0.228 + 0.189i)13-s + (−0.171 − 0.207i)14-s + (0.548 − 0.179i)15-s + (0.242 − 0.0621i)16-s + (−0.163 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0195 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0195 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.0195 - 0.999i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.0195 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.903743 + 0.921623i\)
\(L(\frac12)\) \(\approx\) \(0.903743 + 0.921623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.998 + 0.0627i)T \)
3 \( 1 + (0.904 - 0.425i)T \)
5 \( 1 + (2.21 + 0.275i)T \)
good7 \( 1 + (0.591 + 0.813i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (-0.286 - 4.54i)T + (-10.9 + 1.37i)T^{2} \)
13 \( 1 + (-0.824 - 0.682i)T + (2.43 + 12.7i)T^{2} \)
17 \( 1 + (0.674 - 5.33i)T + (-16.4 - 4.22i)T^{2} \)
19 \( 1 + (0.821 - 1.74i)T + (-12.1 - 14.6i)T^{2} \)
23 \( 1 + (-2.62 - 6.62i)T + (-16.7 + 15.7i)T^{2} \)
29 \( 1 + (-2.35 + 2.21i)T + (1.82 - 28.9i)T^{2} \)
31 \( 1 + (9.46 + 1.19i)T + (30.0 + 7.70i)T^{2} \)
37 \( 1 + (-1.94 - 7.56i)T + (-32.4 + 17.8i)T^{2} \)
41 \( 1 + (2.47 + 0.980i)T + (29.8 + 28.0i)T^{2} \)
43 \( 1 + (6.89 + 2.23i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (-9.48 - 1.81i)T + (43.6 + 17.3i)T^{2} \)
53 \( 1 + (4.40 - 8.01i)T + (-28.3 - 44.7i)T^{2} \)
59 \( 1 + (-2.04 - 3.22i)T + (-25.1 + 53.3i)T^{2} \)
61 \( 1 + (1.90 - 0.753i)T + (44.4 - 41.7i)T^{2} \)
67 \( 1 + (-2.00 + 2.14i)T + (-4.20 - 66.8i)T^{2} \)
71 \( 1 + (-1.73 + 9.09i)T + (-66.0 - 26.1i)T^{2} \)
73 \( 1 + (9.38 + 5.95i)T + (31.0 + 66.0i)T^{2} \)
79 \( 1 + (-4.97 - 10.5i)T + (-50.3 + 60.8i)T^{2} \)
83 \( 1 + (0.234 + 0.110i)T + (52.9 + 63.9i)T^{2} \)
89 \( 1 + (-6.82 + 10.7i)T + (-37.8 - 80.5i)T^{2} \)
97 \( 1 + (7.67 + 8.17i)T + (-6.09 + 96.8i)T^{2} \)
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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

−10.70962426165917120249020855246, −9.982998457077334234524736195641, −8.885830295422833388547612632132, −7.68325003544516519507433241836, −7.06276871571168849559325906795, −6.08744789976782600303422102111, −4.98163980752278540288355896025, −4.15371998329272197353591861767, −3.48470044532690766437005707099, −1.65802533456635095100296625273, 0.56982179909811286013111125898, 2.69562650920500778188334311000, 3.60994393037074767435938549488, 4.75459575924032869858812384961, 5.61045697859526614740850124381, 6.61735620250901471362198664105, 7.26300155916852669847471239628, 8.351251877406243530179028199976, 9.102399031543188997810749975371, 10.65805221246044327941457619650

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