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Properties

Label 2-750-125.94-c1-0-12
Degree $2$
Conductor $750$
Sign $0.608 + 0.793i$
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 + (0.942 − 2.02i)5-s + (0.876 − 0.481i)6-s + (0.754 + 1.03i)7-s + (−0.982 + 0.187i)8-s + (0.637 − 0.770i)9-s + (−0.813 + 2.08i)10-s + (−0.188 − 2.99i)11-s + (−0.844 + 0.535i)12-s + (2.49 + 2.06i)13-s + (−0.818 − 0.989i)14-s + (0.0100 + 2.23i)15-s + (0.968 − 0.248i)16-s + (0.0223 − 0.176i)17-s + ⋯
L(s)  = 1  + (−0.705 + 0.0443i)2-s + (−0.522 + 0.245i)3-s + (0.496 − 0.0626i)4-s + (0.421 − 0.906i)5-s + (0.357 − 0.196i)6-s + (0.285 + 0.392i)7-s + (−0.347 + 0.0662i)8-s + (0.212 − 0.256i)9-s + (−0.257 + 0.658i)10-s + (−0.0568 − 0.902i)11-s + (−0.243 + 0.154i)12-s + (0.691 + 0.571i)13-s + (−0.218 − 0.264i)14-s + (0.00259 + 0.577i)15-s + (0.242 − 0.0621i)16-s + (0.00542 − 0.0429i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.608 + 0.793i$
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: \((2,\ 750,\ (\ :1/2),\ 0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.880982 - 0.434531i\)
\(L(\frac12)\) \(\approx\) \(0.880982 - 0.434531i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.942 + 2.02i)T \)
good7 \( 1 + (-0.754 - 1.03i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (0.188 + 2.99i)T + (-10.9 + 1.37i)T^{2} \)
13 \( 1 + (-2.49 - 2.06i)T + (2.43 + 12.7i)T^{2} \)
17 \( 1 + (-0.0223 + 0.176i)T + (-16.4 - 4.22i)T^{2} \)
19 \( 1 + (-0.423 + 0.900i)T + (-12.1 - 14.6i)T^{2} \)
23 \( 1 + (0.206 + 0.521i)T + (-16.7 + 15.7i)T^{2} \)
29 \( 1 + (-3.98 + 3.73i)T + (1.82 - 28.9i)T^{2} \)
31 \( 1 + (6.83 + 0.863i)T + (30.0 + 7.70i)T^{2} \)
37 \( 1 + (-0.0430 - 0.167i)T + (-32.4 + 17.8i)T^{2} \)
41 \( 1 + (-1.26 - 0.502i)T + (29.8 + 28.0i)T^{2} \)
43 \( 1 + (-5.01 - 1.63i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.29 - 0.438i)T + (43.6 + 17.3i)T^{2} \)
53 \( 1 + (-6.33 + 11.5i)T + (-28.3 - 44.7i)T^{2} \)
59 \( 1 + (2.88 + 4.54i)T + (-25.1 + 53.3i)T^{2} \)
61 \( 1 + (-5.82 + 2.30i)T + (44.4 - 41.7i)T^{2} \)
67 \( 1 + (0.122 - 0.129i)T + (-4.20 - 66.8i)T^{2} \)
71 \( 1 + (-2.32 + 12.1i)T + (-66.0 - 26.1i)T^{2} \)
73 \( 1 + (-4.91 - 3.11i)T + (31.0 + 66.0i)T^{2} \)
79 \( 1 + (-1.95 - 4.14i)T + (-50.3 + 60.8i)T^{2} \)
83 \( 1 + (-3.73 - 1.75i)T + (52.9 + 63.9i)T^{2} \)
89 \( 1 + (-7.46 + 11.7i)T + (-37.8 - 80.5i)T^{2} \)
97 \( 1 + (-4.28 - 4.55i)T + (-6.09 + 96.8i)T^{2} \)
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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

−10.14128611759442384850989116242, −9.245587695650009174275243942700, −8.704746021274078809624490211863, −7.913596595271262369283206324470, −6.58330845367581259369803272315, −5.82551881926188642383597859941, −5.02613906098911214581033520586, −3.77740683773143841881542116758, −2.10759021736358533478994257293, −0.76070713377582092704462685419, 1.32720167944181808672238717536, 2.57007675294274803621946976413, 3.91765599785439478976356633688, 5.36993137798837824650542728914, 6.23577375947191826942385296644, 7.17876241158932862415017344097, 7.63740607408056687111225475538, 8.844915905786396986064608329060, 9.807322645301758538027407423755, 10.67119793282550937555361517410

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