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Properties

Label 2-750-125.104-c1-0-11
Degree $2$
Conductor $750$
Sign $0.171 - 0.985i$
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 + (1.61 + 1.54i)5-s + (0.425 + 0.904i)6-s + (0.699 − 0.962i)7-s + (−0.684 − 0.728i)8-s + (0.929 − 0.368i)9-s + (−1.09 + 1.94i)10-s + (−0.289 + 0.0743i)11-s + (−0.770 + 0.637i)12-s + (1.45 + 3.68i)13-s + (1.10 + 0.438i)14-s + (1.87 + 1.21i)15-s + (0.535 − 0.844i)16-s + (−0.233 + 0.425i)17-s + ⋯
L(s)  = 1  + (0.175 + 0.684i)2-s + (0.567 − 0.108i)3-s + (−0.438 + 0.240i)4-s + (0.721 + 0.692i)5-s + (0.173 + 0.369i)6-s + (0.264 − 0.363i)7-s + (−0.242 − 0.257i)8-s + (0.309 − 0.122i)9-s + (−0.347 + 0.615i)10-s + (−0.0872 + 0.0224i)11-s + (−0.222 + 0.184i)12-s + (0.404 + 1.02i)13-s + (0.295 + 0.117i)14-s + (0.483 + 0.314i)15-s + (0.133 − 0.211i)16-s + (−0.0566 + 0.103i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.171 - 0.985i$
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: \((2,\ 750,\ (\ :1/2),\ 0.171 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72026 + 1.44688i\)
\(L(\frac12)\) \(\approx\) \(1.72026 + 1.44688i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 - 1.54i)T \)
good7 \( 1 + (-0.699 + 0.962i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.289 - 0.0743i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (-1.45 - 3.68i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (0.233 - 0.425i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (0.106 - 0.557i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-3.48 - 0.219i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (-8.38 - 1.05i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (5.20 + 2.86i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (5.30 + 3.36i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (-0.0193 - 0.307i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (3.49 - 1.13i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-2.23 + 2.38i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (6.75 + 3.17i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (4.63 + 5.59i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (0.0466 - 0.741i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-0.320 - 2.53i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (5.11 + 4.80i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (-7.91 - 6.54i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (0.548 + 2.87i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (-14.6 - 2.80i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-2.87 + 3.47i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (-1.39 + 11.0i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.45544747728943179310896035809, −9.484702491186116849674818731762, −8.834450525614736222651817782561, −7.84319528632880876983580506528, −6.94230693141645324421814550384, −6.40577767315697766364546853109, −5.24766183759258392757269800764, −4.14403943182663221621116377629, −3.05759696393550105441341353890, −1.71931091962474052257032125907, 1.18383801383032894739549105737, 2.42055243355390947821431761157, 3.40864509635062056801798622511, 4.76195720392105021906710095900, 5.37117266033850261883295229718, 6.50170885537283779359481578871, 7.932178596424057788776935808284, 8.691870402196424830949932110186, 9.258801831286796974536470728472, 10.25184990179816680558064529939

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