Properties

Label 2-756-84.23-c1-0-20
Degree $2$
Conductor $756$
Sign $0.655 - 0.755i$
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  + (1.38 + 0.290i)2-s + (1.83 + 0.803i)4-s + (−2.75 − 1.59i)5-s + (−0.944 + 2.47i)7-s + (2.30 + 1.64i)8-s + (−3.35 − 3.00i)10-s + (1.24 + 2.16i)11-s + 6.61·13-s + (−2.02 + 3.14i)14-s + (2.70 + 2.94i)16-s + (2.67 − 1.54i)17-s + (4.40 + 2.54i)19-s + (−3.77 − 5.13i)20-s + (1.10 + 3.35i)22-s + (−2.53 + 4.39i)23-s + ⋯
L(s)  = 1  + (0.978 + 0.205i)2-s + (0.915 + 0.401i)4-s + (−1.23 − 0.712i)5-s + (−0.356 + 0.934i)7-s + (0.813 + 0.581i)8-s + (−1.06 − 0.950i)10-s + (0.376 + 0.651i)11-s + 1.83·13-s + (−0.541 + 0.841i)14-s + (0.677 + 0.735i)16-s + (0.649 − 0.374i)17-s + (1.01 + 0.583i)19-s + (−0.843 − 1.14i)20-s + (0.234 + 0.715i)22-s + (−0.529 + 0.916i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.655 - 0.755i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.655 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22662 + 1.01550i\)
\(L(\frac12)\) \(\approx\) \(2.22662 + 1.01550i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.290i)T \)
3 \( 1 \)
7 \( 1 + (0.944 - 2.47i)T \)
good5 \( 1 + (2.75 + 1.59i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.24 - 2.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.61T + 13T^{2} \)
17 \( 1 + (-2.67 + 1.54i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.40 - 2.54i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.53 - 4.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.59iT - 29T^{2} \)
31 \( 1 + (7.31 - 4.22i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.357 - 0.619i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12.2iT - 41T^{2} \)
43 \( 1 + 3.72iT - 43T^{2} \)
47 \( 1 + (-2.51 + 4.35i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.21 - 4.16i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.36 + 9.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.997 + 1.72i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.00277 - 0.00160i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + (-0.957 - 1.65i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.11 - 3.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.93T + 83T^{2} \)
89 \( 1 + (0.482 + 0.278i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.127T + 97T^{2} \)
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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

−10.83142958087800048868387165861, −9.442954463779181082518602052352, −8.561806739404388293563144029853, −7.80503104927118378158275469135, −6.93487062132484912354391281781, −5.76451668138371984370699697480, −5.16488059663790687470996831760, −3.78868582829250742420714417317, −3.47448501887130927421995925666, −1.61324995025645879956972378108, 1.06115730518947303035949287163, 3.15464857508528308504928428749, 3.64242651231172826658749938108, 4.38772221946942792723655202792, 5.93558047092979111361015389649, 6.55698648993908169834836098085, 7.51530810220588954296754577944, 8.172678099750389313835799072028, 9.608517676099800127467160792156, 10.74690477670629597788278926430

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