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Properties

Label 2-756-36.23-c1-0-29
Degree $2$
Conductor $756$
Sign $-0.834 - 0.551i$
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  + (−0.726 − 1.21i)2-s + (−0.944 + 1.76i)4-s + (−0.256 − 0.148i)5-s + (−0.866 + 0.5i)7-s + (2.82 − 0.135i)8-s + (0.00667 + 0.418i)10-s + (−2.21 − 3.83i)11-s + (−0.227 + 0.393i)13-s + (1.23 + 0.687i)14-s + (−2.21 − 3.32i)16-s + 6.53i·17-s − 1.47i·19-s + (0.503 − 0.312i)20-s + (−3.04 + 5.47i)22-s + (2.09 − 3.62i)23-s + ⋯
L(s)  = 1  + (−0.513 − 0.857i)2-s + (−0.472 + 0.881i)4-s + (−0.114 − 0.0662i)5-s + (−0.327 + 0.188i)7-s + (0.998 − 0.0477i)8-s + (0.00210 + 0.132i)10-s + (−0.668 − 1.15i)11-s + (−0.0630 + 0.109i)13-s + (0.330 + 0.183i)14-s + (−0.554 − 0.832i)16-s + 1.58i·17-s − 0.338i·19-s + (0.112 − 0.0698i)20-s + (−0.649 + 1.16i)22-s + (0.436 − 0.755i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.834 - 0.551i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.834 - 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0458787 + 0.152704i\)
\(L(\frac12)\) \(\approx\) \(0.0458787 + 0.152704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.726 + 1.21i)T \)
3 \( 1 \)
7 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (0.256 + 0.148i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.21 + 3.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.227 - 0.393i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.53iT - 17T^{2} \)
19 \( 1 + 1.47iT - 19T^{2} \)
23 \( 1 + (-2.09 + 3.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.71 - 3.30i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.73 + 3.31i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.53T + 37T^{2} \)
41 \( 1 + (1.57 + 0.906i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.80 - 5.08i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.32 + 5.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.22iT - 53T^{2} \)
59 \( 1 + (6.57 - 11.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.86 - 3.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.941 - 0.543i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.66T + 71T^{2} \)
73 \( 1 - 3.76T + 73T^{2} \)
79 \( 1 + (-4.13 + 2.38i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.93 + 8.54i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.2iT - 89T^{2} \)
97 \( 1 + (-5.52 - 9.57i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.03965481147035648309134801625, −8.850274710500324682487506023107, −8.476842876399993351712879581463, −7.51544573331793553913227043972, −6.32443733950930569156895701843, −5.23332304314576691170426671125, −3.94180896769483732101601094860, −3.11322184413515179488412869481, −1.85078555902165336850367397724, −0.092579773453221573454321182169, 1.84603101332692541544727490025, 3.51631535381383255303854391594, 4.92815236415348164050180043168, 5.45082282642984269316559360323, 6.84769243894628311480128071447, 7.31667404314519970600983256718, 8.061942436312371904505069059140, 9.399457591298850207889315582757, 9.604480310219266530920038458997, 10.62071190967607327611752187849

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