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Properties

Label 2-750-125.9-c1-0-15
Degree $2$
Conductor $750$
Sign $0.524 + 0.851i$
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  + (−0.904 − 0.425i)2-s + (−0.998 + 0.0627i)3-s + (0.637 + 0.770i)4-s + (2.16 − 0.564i)5-s + (0.929 + 0.368i)6-s + (2.48 − 0.808i)7-s + (−0.248 − 0.968i)8-s + (0.992 − 0.125i)9-s + (−2.19 − 0.410i)10-s + (1.27 − 2.71i)11-s + (−0.684 − 0.728i)12-s + (−0.257 − 2.04i)13-s + (−2.59 − 0.327i)14-s + (−2.12 + 0.699i)15-s + (−0.187 + 0.982i)16-s + (1.79 + 1.48i)17-s + ⋯
L(s)  = 1  + (−0.639 − 0.301i)2-s + (−0.576 + 0.0362i)3-s + (0.318 + 0.385i)4-s + (0.967 − 0.252i)5-s + (0.379 + 0.150i)6-s + (0.940 − 0.305i)7-s + (−0.0879 − 0.342i)8-s + (0.330 − 0.0417i)9-s + (−0.695 − 0.129i)10-s + (0.384 − 0.817i)11-s + (−0.197 − 0.210i)12-s + (−0.0715 − 0.566i)13-s + (−0.693 − 0.0876i)14-s + (−0.548 + 0.180i)15-s + (−0.0468 + 0.245i)16-s + (0.436 + 0.360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.524 + 0.851i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.524 + 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08443 - 0.605614i\)
\(L(\frac12)\) \(\approx\) \(1.08443 - 0.605614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.904 + 0.425i)T \)
3 \( 1 + (0.998 - 0.0627i)T \)
5 \( 1 + (-2.16 + 0.564i)T \)
good7 \( 1 + (-2.48 + 0.808i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (-1.27 + 2.71i)T + (-7.01 - 8.47i)T^{2} \)
13 \( 1 + (0.257 + 2.04i)T + (-12.5 + 3.23i)T^{2} \)
17 \( 1 + (-1.79 - 1.48i)T + (3.18 + 16.6i)T^{2} \)
19 \( 1 + (0.279 - 4.43i)T + (-18.8 - 2.38i)T^{2} \)
23 \( 1 + (3.36 + 6.12i)T + (-12.3 + 19.4i)T^{2} \)
29 \( 1 + (0.548 - 0.864i)T + (-12.3 - 26.2i)T^{2} \)
31 \( 1 + (3.68 - 4.45i)T + (-5.80 - 30.4i)T^{2} \)
37 \( 1 + (-3.80 - 0.725i)T + (34.4 + 13.6i)T^{2} \)
41 \( 1 + (-4.62 - 2.54i)T + (21.9 + 34.6i)T^{2} \)
43 \( 1 + (-0.314 - 0.433i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (-1.56 + 6.08i)T + (-41.1 - 22.6i)T^{2} \)
53 \( 1 + (4.25 + 10.7i)T + (-38.6 + 36.2i)T^{2} \)
59 \( 1 + (-9.96 + 9.36i)T + (3.70 - 58.8i)T^{2} \)
61 \( 1 + (-1.89 + 1.04i)T + (32.6 - 51.5i)T^{2} \)
67 \( 1 + (5.19 - 3.29i)T + (28.5 - 60.6i)T^{2} \)
71 \( 1 + (4.84 + 1.24i)T + (62.2 + 34.2i)T^{2} \)
73 \( 1 + (-7.00 + 7.45i)T + (-4.58 - 72.8i)T^{2} \)
79 \( 1 + (0.0625 + 0.994i)T + (-78.3 + 9.90i)T^{2} \)
83 \( 1 + (-0.824 - 0.0518i)T + (82.3 + 10.4i)T^{2} \)
89 \( 1 + (-12.1 - 11.4i)T + (5.58 + 88.8i)T^{2} \)
97 \( 1 + (-7.01 - 4.45i)T + (41.3 + 87.7i)T^{2} \)
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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

−10.37884205264348653450956894745, −9.511645134009548943909612381871, −8.459755283391273331661134533964, −7.925268000885901441385291150802, −6.60308549343311773230910398581, −5.83909047680433054853546759281, −4.91592360631009150320215315213, −3.62791728844010419252172492240, −2.04772448203023247366863043869, −0.966722881062727289855222492871, 1.42714058082066707073989284643, 2.38063227831796435468865444940, 4.36808428667088362040686207323, 5.37168518827765036122854393979, 6.07045579288070125920801356011, 7.11272605428903451114102644147, 7.71116949805911367008225399671, 9.104414488890567451751000646793, 9.487753183689408312431969580495, 10.39349406025430108040751358899

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