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Properties

Label 2-756-84.11-c1-0-63
Degree $2$
Conductor $756$
Sign $-0.548 - 0.836i$
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  + (0.687 − 1.23i)2-s + (−1.05 − 1.70i)4-s + (−0.303 + 0.175i)5-s + (−2.63 + 0.234i)7-s + (−2.82 + 0.132i)8-s + (0.00772 + 0.495i)10-s + (−0.356 + 0.617i)11-s + 0.127·13-s + (−1.52 + 3.41i)14-s + (−1.78 + 3.58i)16-s + (−5.30 − 3.06i)17-s + (−2.91 + 1.68i)19-s + (0.617 + 0.331i)20-s + (0.517 + 0.864i)22-s + (−2.38 − 4.13i)23-s + ⋯
L(s)  = 1  + (0.486 − 0.873i)2-s + (−0.526 − 0.850i)4-s + (−0.135 + 0.0783i)5-s + (−0.996 + 0.0884i)7-s + (−0.998 + 0.0467i)8-s + (0.00244 + 0.156i)10-s + (−0.107 + 0.186i)11-s + 0.0353·13-s + (−0.407 + 0.913i)14-s + (−0.445 + 0.895i)16-s + (−1.28 − 0.742i)17-s + (−0.669 + 0.386i)19-s + (0.138 + 0.0740i)20-s + (0.110 + 0.184i)22-s + (−0.497 − 0.862i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.548 - 0.836i$
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: \((2,\ 756,\ (\ :1/2),\ -0.548 - 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0994721 + 0.184092i\)
\(L(\frac12)\) \(\approx\) \(0.0994721 + 0.184092i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.687 + 1.23i)T \)
3 \( 1 \)
7 \( 1 + (2.63 - 0.234i)T \)
good5 \( 1 + (0.303 - 0.175i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.356 - 0.617i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.127T + 13T^{2} \)
17 \( 1 + (5.30 + 3.06i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.91 - 1.68i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.38 + 4.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.39iT - 29T^{2} \)
31 \( 1 + (0.00202 + 0.00116i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.17 - 3.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.22iT - 41T^{2} \)
43 \( 1 - 7.55iT - 43T^{2} \)
47 \( 1 + (1.33 + 2.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.1 + 5.86i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.13 + 8.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.83 + 4.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.79 + 4.50i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.61T + 71T^{2} \)
73 \( 1 + (2.15 - 3.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.92 + 1.10i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.12T + 83T^{2} \)
89 \( 1 + (7.79 - 4.49i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−9.836958481937967826863528572572, −9.250137592316889374079043480700, −8.299674244483105204210179651798, −6.86118765989705419815200799206, −6.20795815270565505941833369106, −5.05155091342515955448597313594, −4.08668106453768293985093633040, −3.09992864628176551458947456450, −2.03985666402771018734031362909, −0.084009004328161771787828099469, 2.55133137334395765918705383952, 3.80795900467121703851312212430, 4.50092582676263382781778517039, 5.89642313065537864619096920267, 6.36448974314717745184199447446, 7.31608503478607350096582441651, 8.250680014500491756308117356035, 9.020348062817317498338118932565, 9.856248696399495684611176062730, 10.92973809148156429637110766007

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