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Properties

Label 2-750-125.104-c1-0-12
Degree $2$
Conductor $750$
Sign $0.803 - 0.594i$
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  + (0.248 + 0.968i)2-s + (0.982 − 0.187i)3-s + (−0.876 + 0.481i)4-s + (−2.23 + 0.0625i)5-s + (0.425 + 0.904i)6-s + (1.22 − 1.68i)7-s + (−0.684 − 0.728i)8-s + (0.929 − 0.368i)9-s + (−0.616 − 2.14i)10-s + (3.55 − 0.911i)11-s + (−0.770 + 0.637i)12-s + (1.39 + 3.53i)13-s + (1.93 + 0.767i)14-s + (−2.18 + 0.480i)15-s + (0.535 − 0.844i)16-s + (2.60 − 4.74i)17-s + ⋯
L(s)  = 1  + (0.175 + 0.684i)2-s + (0.567 − 0.108i)3-s + (−0.438 + 0.240i)4-s + (−0.999 + 0.0279i)5-s + (0.173 + 0.369i)6-s + (0.463 − 0.637i)7-s + (−0.242 − 0.257i)8-s + (0.309 − 0.122i)9-s + (−0.194 − 0.679i)10-s + (1.07 − 0.274i)11-s + (−0.222 + 0.184i)12-s + (0.388 + 0.980i)13-s + (0.518 + 0.205i)14-s + (−0.563 + 0.124i)15-s + (0.133 − 0.211i)16-s + (0.632 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.803 - 0.594i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.803 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77520 + 0.585187i\)
\(L(\frac12)\) \(\approx\) \(1.77520 + 0.585187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.248 - 0.968i)T \)
3 \( 1 + (-0.982 + 0.187i)T \)
5 \( 1 + (2.23 - 0.0625i)T \)
good7 \( 1 + (-1.22 + 1.68i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (-3.55 + 0.911i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (-1.39 - 3.53i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (-2.60 + 4.74i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (0.648 - 3.39i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-4.02 - 0.253i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (2.86 + 0.361i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (-4.44 - 2.44i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (-8.26 - 5.24i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (0.619 + 9.84i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (8.35 - 2.71i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (0.162 - 0.173i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (-2.97 - 1.40i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (-3.40 - 4.11i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (-0.0306 + 0.487i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (0.575 + 4.55i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (3.12 + 2.93i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (9.83 + 8.13i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (1.68 + 8.84i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (10.4 + 1.98i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-3.99 + 4.82i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (-1.50 + 11.9i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.36777729448993106409574876109, −9.231879502552697732462443627220, −8.635881497460087920548747090048, −7.69972712779098398373059183865, −7.15012013847154958360386381996, −6.28702702507499572028180307732, −4.79708642778404471651584292808, −4.06334564103051337223208654905, −3.21974530875550288195369062891, −1.18592238747629477671265468222, 1.22605925476946573102090568751, 2.72301691687903838756025124737, 3.69114888591071192860323077230, 4.48094249663794176752514234585, 5.60354195966540474201330863076, 6.86127194951096642644927376666, 8.071250667725618896200574206368, 8.499413068152441485363599449601, 9.394789982315090883395388388008, 10.33927570624675118458182240917

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