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Properties

Label 2-756-63.25-c1-0-1
Degree $2$
Conductor $756$
Sign $0.361 - 0.932i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (0.483 − 0.838i)5-s + (−1.52 + 2.16i)7-s + (0.364 + 0.630i)11-s + (1.81 + 3.13i)13-s + (−3.49 + 6.06i)17-s + (−0.348 − 0.602i)19-s + (3.21 − 5.57i)23-s + (2.03 + 3.51i)25-s + (−3.34 + 5.79i)29-s + 9.16·31-s + (1.07 + 2.32i)35-s + (0.854 + 1.48i)37-s + (3.62 + 6.27i)41-s + (−0.348 + 0.602i)43-s − 7.66·47-s + ⋯
L(s)  = 1  + (0.216 − 0.374i)5-s + (−0.574 + 0.818i)7-s + (0.109 + 0.190i)11-s + (0.502 + 0.869i)13-s + (−0.848 + 1.47i)17-s + (−0.0798 − 0.138i)19-s + (0.671 − 1.16i)23-s + (0.406 + 0.703i)25-s + (−0.621 + 1.07i)29-s + 1.64·31-s + (0.182 + 0.392i)35-s + (0.140 + 0.243i)37-s + (0.566 + 0.980i)41-s + (−0.0530 + 0.0919i)43-s − 1.11·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.361 - 0.932i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.361 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10493 + 0.756409i\)
\(L(\frac12)\) \(\approx\) \(1.10493 + 0.756409i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (1.52 - 2.16i)T \)
good5 \( 1 + (-0.483 + 0.838i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.364 - 0.630i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.81 - 3.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.49 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.348 + 0.602i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.21 + 5.57i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.34 - 5.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.16T + 31T^{2} \)
37 \( 1 + (-0.854 - 1.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.62 - 6.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.348 - 0.602i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.66T + 47T^{2} \)
53 \( 1 + (2.05 - 3.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.77T + 59T^{2} \)
61 \( 1 - 4.92T + 61T^{2} \)
67 \( 1 + 5.82T + 67T^{2} \)
71 \( 1 + 0.304T + 71T^{2} \)
73 \( 1 + (-5.33 + 9.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 3.23T + 79T^{2} \)
83 \( 1 + (0.618 - 1.07i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.78 - 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.32 + 2.30i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.53166572148747460447174384254, −9.453216953807723685254559110237, −8.866346128279334029068664843639, −8.209220498742834713125541394779, −6.66424720774336444674384134320, −6.32812321190153909478208546548, −5.10081169969480527949077934780, −4.14657843700186408526615231387, −2.85314601324382354080623649677, −1.58838110706688331661265815806, 0.71121543595794294257056275114, 2.59914600701656372862567924273, 3.55510227415831250181493433574, 4.68821161866018996875323037657, 5.85676486693544809128584019483, 6.73099167852996252861061777631, 7.45809595815053261942158944678, 8.461536385928531132829603490665, 9.531363536930822124363824618665, 10.10275400321335855433515841733

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