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Properties

Label 2-756-84.11-c1-0-12
Degree $2$
Conductor $756$
Sign $0.335 - 0.942i$
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  + (−1.04 − 0.948i)2-s + (0.199 + 1.99i)4-s + (−0.479 + 0.277i)5-s + (2.45 + 0.976i)7-s + (1.67 − 2.27i)8-s + (0.766 + 0.164i)10-s + (−2.96 + 5.14i)11-s + 3.20·13-s + (−1.65 − 3.35i)14-s + (−3.92 + 0.793i)16-s + (−2.48 − 1.43i)17-s + (−3.43 + 1.98i)19-s + (−0.646 − 0.899i)20-s + (7.99 − 2.57i)22-s + (0.145 + 0.251i)23-s + ⋯
L(s)  = 1  + (−0.741 − 0.670i)2-s + (0.0996 + 0.995i)4-s + (−0.214 + 0.123i)5-s + (0.929 + 0.369i)7-s + (0.593 − 0.804i)8-s + (0.242 + 0.0521i)10-s + (−0.895 + 1.55i)11-s + 0.888·13-s + (−0.441 − 0.897i)14-s + (−0.980 + 0.198i)16-s + (−0.602 − 0.348i)17-s + (−0.788 + 0.455i)19-s + (−0.144 − 0.201i)20-s + (1.70 − 0.549i)22-s + (0.0303 + 0.0525i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.335 - 0.942i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.335 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.655660 + 0.462704i\)
\(L(\frac12)\) \(\approx\) \(0.655660 + 0.462704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.04 + 0.948i)T \)
3 \( 1 \)
7 \( 1 + (-2.45 - 0.976i)T \)
good5 \( 1 + (0.479 - 0.277i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.96 - 5.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.20T + 13T^{2} \)
17 \( 1 + (2.48 + 1.43i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.43 - 1.98i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.145 - 0.251i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.13iT - 29T^{2} \)
31 \( 1 + (5.96 + 3.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.20 - 2.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.27iT - 41T^{2} \)
43 \( 1 - 8.31iT - 43T^{2} \)
47 \( 1 + (-6.19 - 10.7i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.21 + 2.43i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.24 - 10.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.305 + 0.529i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.70 - 2.13i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.00T + 71T^{2} \)
73 \( 1 + (-6.28 + 10.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.20 - 3.00i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.675T + 83T^{2} \)
89 \( 1 + (-11.1 + 6.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.00T + 97T^{2} \)
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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

−10.63605226091132098728324010806, −9.640312173329714379813535945665, −8.935317736826896375685068771054, −7.82265123797479284595758951952, −7.59775318688346770416726799993, −6.23869944123343760171877204741, −4.83356755282194862612554077003, −4.02471774985429535611874852124, −2.54309127155355921148498707379, −1.65868855025447165446303680379, 0.52130931027752848336143411366, 2.04681099376066692625816759810, 3.78280583745025736268239044355, 5.02469035041487472379153265225, 5.81724426535426084122016071997, 6.78416681153267185366025436089, 7.78782007494517292735549417278, 8.594903505562618980274556145562, 8.781102916079255364319129012322, 10.35440943588813788939540507518

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