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Properties

Label 2-750-25.9-c1-0-8
Degree $2$
Conductor $750$
Sign $0.778 + 0.628i$
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  + (−0.587 + 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.309 + 0.951i)6-s − 2.61i·7-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (−2.92 − 2.12i)11-s + (−0.587 − 0.809i)12-s + (3.80 + 5.23i)13-s + (2.11 + 1.53i)14-s + (−0.809 + 0.587i)16-s + (−1.17 − 0.381i)17-s + 0.999i·18-s + (1.76 − 5.42i)19-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (0.549 − 0.178i)3-s + (−0.154 − 0.475i)4-s + (−0.126 + 0.388i)6-s − 0.989i·7-s + (0.336 + 0.109i)8-s + (0.269 − 0.195i)9-s + (−0.882 − 0.641i)11-s + (−0.169 − 0.233i)12-s + (1.05 + 1.45i)13-s + (0.566 + 0.411i)14-s + (−0.202 + 0.146i)16-s + (−0.285 − 0.0926i)17-s + 0.235i·18-s + (0.404 − 1.24i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.778 + 0.628i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.778 + 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28952 - 0.455600i\)
\(L(\frac12)\) \(\approx\) \(1.28952 - 0.455600i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 \)
good7 \( 1 + 2.61iT - 7T^{2} \)
11 \( 1 + (2.92 + 2.12i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.80 - 5.23i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.17 + 0.381i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.76 + 5.42i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.62 + 3.61i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.61 + 8.05i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.04 + 6.29i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.70 - 6.47i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.61 - 3.35i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 7.70iT - 43T^{2} \)
47 \( 1 + (-1.62 + 0.527i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.98 + 0.645i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.92 - 2.12i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.23 + 1.62i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.45 - 0.472i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.70 - 5.25i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.07 + 2.85i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.73 - 5.34i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.04 - 0.663i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-2.85 - 2.07i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.21 - 1.04i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.09908914001680626683574680923, −9.225579290241313725287335607648, −8.505744936567859200396282255749, −7.70869110173241888762364487214, −6.87006364271648120430524555328, −6.16632922058012125467512715446, −4.76139326360064988823292180865, −3.87296611964674414446550338785, −2.43001003875025848185115134750, −0.804793743482930416750146955707, 1.57332820834218130369192555829, 2.84116539634868710428113919951, 3.55708058928482161505752480015, 5.05856299813632001207700502815, 5.83110468748523197277077009902, 7.34221832183350182157068938335, 8.114295384210941482123988474012, 8.766693269175112654337077410520, 9.586010699521874938037540844201, 10.46912308168078689141340423685

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