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Properties

Label 2-750-125.94-c1-0-14
Degree $2$
Conductor $750$
Sign $0.332 + 0.942i$
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 + (1.08 + 1.95i)5-s + (0.876 − 0.481i)6-s + (0.161 + 0.221i)7-s + (−0.982 + 0.187i)8-s + (0.637 − 0.770i)9-s + (−1.20 − 1.88i)10-s + (−0.316 − 5.03i)11-s + (−0.844 + 0.535i)12-s + (−4.85 − 4.01i)13-s + (−0.174 − 0.211i)14-s + (−1.81 − 1.30i)15-s + (0.968 − 0.248i)16-s + (0.721 − 5.71i)17-s + ⋯
L(s)  = 1  + (−0.705 + 0.0443i)2-s + (−0.522 + 0.245i)3-s + (0.496 − 0.0626i)4-s + (0.485 + 0.874i)5-s + (0.357 − 0.196i)6-s + (0.0609 + 0.0838i)7-s + (−0.347 + 0.0662i)8-s + (0.212 − 0.256i)9-s + (−0.381 − 0.595i)10-s + (−0.0955 − 1.51i)11-s + (−0.243 + 0.154i)12-s + (−1.34 − 1.11i)13-s + (−0.0467 − 0.0564i)14-s + (−0.468 − 0.337i)15-s + (0.242 − 0.0621i)16-s + (0.175 − 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.332 + 0.942i$
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: \((2,\ 750,\ (\ :1/2),\ 0.332 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.543682 - 0.384662i\)
\(L(\frac12)\) \(\approx\) \(0.543682 - 0.384662i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.08 - 1.95i)T \)
good7 \( 1 + (-0.161 - 0.221i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (0.316 + 5.03i)T + (-10.9 + 1.37i)T^{2} \)
13 \( 1 + (4.85 + 4.01i)T + (2.43 + 12.7i)T^{2} \)
17 \( 1 + (-0.721 + 5.71i)T + (-16.4 - 4.22i)T^{2} \)
19 \( 1 + (3.03 - 6.44i)T + (-12.1 - 14.6i)T^{2} \)
23 \( 1 + (0.824 + 2.08i)T + (-16.7 + 15.7i)T^{2} \)
29 \( 1 + (-6.57 + 6.17i)T + (1.82 - 28.9i)T^{2} \)
31 \( 1 + (-8.18 - 1.03i)T + (30.0 + 7.70i)T^{2} \)
37 \( 1 + (1.32 + 5.16i)T + (-32.4 + 17.8i)T^{2} \)
41 \( 1 + (6.58 + 2.60i)T + (29.8 + 28.0i)T^{2} \)
43 \( 1 + (-1.60 - 0.520i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (8.74 + 1.66i)T + (43.6 + 17.3i)T^{2} \)
53 \( 1 + (2.13 - 3.89i)T + (-28.3 - 44.7i)T^{2} \)
59 \( 1 + (-1.46 - 2.31i)T + (-25.1 + 53.3i)T^{2} \)
61 \( 1 + (0.525 - 0.208i)T + (44.4 - 41.7i)T^{2} \)
67 \( 1 + (-2.18 + 2.32i)T + (-4.20 - 66.8i)T^{2} \)
71 \( 1 + (-2.95 + 15.5i)T + (-66.0 - 26.1i)T^{2} \)
73 \( 1 + (0.955 + 0.606i)T + (31.0 + 66.0i)T^{2} \)
79 \( 1 + (-0.844 - 1.79i)T + (-50.3 + 60.8i)T^{2} \)
83 \( 1 + (-11.0 - 5.22i)T + (52.9 + 63.9i)T^{2} \)
89 \( 1 + (-1.18 + 1.86i)T + (-37.8 - 80.5i)T^{2} \)
97 \( 1 + (5.44 + 5.80i)T + (-6.09 + 96.8i)T^{2} \)
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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

−10.23274415602294825492586974963, −9.645887094502170335534693524123, −8.386944533845950403936869273540, −7.70791758377625482946073034699, −6.59265079772404317662003124640, −5.92887801522937348368627203056, −5.02900658305018461858463474236, −3.32653937663080321990640034238, −2.44062755740947557687098642902, −0.45870789305154810743803929430, 1.45110627575041307451066135907, 2.36268566530541309024630394060, 4.53907695770920950966354991400, 4.93014077280446418795771449754, 6.45807827810824405783126555639, 6.92941759104453615106665113353, 8.041054323612970652342327705297, 8.873869877750042131922494435787, 9.856516385241601332512745052493, 10.14626035945301398934696800424

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