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Properties

Label 2-750-125.96-c1-0-12
Degree $2$
Conductor $750$
Sign $0.786 - 0.618i$
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  + (0.929 + 0.368i)2-s + (0.876 − 0.481i)3-s + (0.728 + 0.684i)4-s + (1.19 + 1.89i)5-s + (0.992 − 0.125i)6-s + (−2.11 − 1.53i)7-s + (0.425 + 0.904i)8-s + (0.535 − 0.844i)9-s + (0.410 + 2.19i)10-s + (3.11 + 1.23i)11-s + (0.968 + 0.248i)12-s + (−0.861 + 1.35i)13-s + (−1.40 − 2.20i)14-s + (1.95 + 1.08i)15-s + (0.0627 + 0.998i)16-s + (0.0898 − 0.0843i)17-s + ⋯
L(s)  = 1  + (0.657 + 0.260i)2-s + (0.505 − 0.278i)3-s + (0.364 + 0.342i)4-s + (0.532 + 0.846i)5-s + (0.405 − 0.0511i)6-s + (−0.799 − 0.580i)7-s + (0.150 + 0.319i)8-s + (0.178 − 0.281i)9-s + (0.129 + 0.695i)10-s + (0.940 + 0.372i)11-s + (0.279 + 0.0717i)12-s + (−0.239 + 0.376i)13-s + (−0.374 − 0.589i)14-s + (0.504 + 0.280i)15-s + (0.0156 + 0.249i)16-s + (0.0217 − 0.0204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.786 - 0.618i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.786 - 0.618i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.66406 + 0.922046i\)
\(L(\frac12)\) \(\approx\) \(2.66406 + 0.922046i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.929 - 0.368i)T \)
3 \( 1 + (-0.876 + 0.481i)T \)
5 \( 1 + (-1.19 - 1.89i)T \)
good7 \( 1 + (2.11 + 1.53i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-3.11 - 1.23i)T + (8.01 + 7.53i)T^{2} \)
13 \( 1 + (0.861 - 1.35i)T + (-5.53 - 11.7i)T^{2} \)
17 \( 1 + (-0.0898 + 0.0843i)T + (1.06 - 16.9i)T^{2} \)
19 \( 1 + (-6.70 - 3.68i)T + (10.1 + 16.0i)T^{2} \)
23 \( 1 + (-5.61 + 6.78i)T + (-4.30 - 22.5i)T^{2} \)
29 \( 1 + (-0.661 - 3.46i)T + (-26.9 + 10.6i)T^{2} \)
31 \( 1 + (7.59 - 7.12i)T + (1.94 - 30.9i)T^{2} \)
37 \( 1 + (0.552 + 8.78i)T + (-36.7 + 4.63i)T^{2} \)
41 \( 1 + (6.77 + 8.18i)T + (-7.68 + 40.2i)T^{2} \)
43 \( 1 + (-0.0707 - 0.217i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (2.97 - 6.32i)T + (-29.9 - 36.2i)T^{2} \)
53 \( 1 + (7.18 + 0.907i)T + (51.3 + 13.1i)T^{2} \)
59 \( 1 + (4.26 + 1.09i)T + (51.7 + 28.4i)T^{2} \)
61 \( 1 + (-6.81 + 8.23i)T + (-11.4 - 59.9i)T^{2} \)
67 \( 1 + (-1.22 + 6.40i)T + (-62.2 - 24.6i)T^{2} \)
71 \( 1 + (3.74 - 7.95i)T + (-45.2 - 54.7i)T^{2} \)
73 \( 1 + (2.19 - 0.564i)T + (63.9 - 35.1i)T^{2} \)
79 \( 1 + (-4.99 + 2.74i)T + (42.3 - 66.7i)T^{2} \)
83 \( 1 + (10.7 + 5.89i)T + (44.4 + 70.0i)T^{2} \)
89 \( 1 + (-9.97 + 2.56i)T + (77.9 - 42.8i)T^{2} \)
97 \( 1 + (0.851 + 4.46i)T + (-90.1 + 35.7i)T^{2} \)
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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

−10.41677040308674680894165150038, −9.582686953444024355328050335204, −8.833227779648461799397008499131, −7.30776759518592037160176545617, −7.03209207457434598621828965929, −6.25307470673804477782575645004, −5.08027425042960973186403831801, −3.68259421131510077244264453324, −3.13551875534870251506432054867, −1.73195923672779095647503405457, 1.33300083243699260902704010793, 2.81639199035164848114532609511, 3.60242917239140135826466979966, 4.90214801716517173075634351786, 5.57926938524130337532298321891, 6.54105540608136492448145731316, 7.67128817869754475178976269713, 8.904278864331927542037086694512, 9.470935703031324915263668854732, 9.938502683867421665006902435479

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