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Properties

Label 2-756-63.16-c1-0-0
Degree $2$
Conductor $756$
Sign $-0.831 - 0.556i$
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  − 0.967·5-s + (−1.11 + 2.39i)7-s − 0.728·11-s + (1.81 + 3.13i)13-s + (−3.49 − 6.06i)17-s + (−0.348 + 0.602i)19-s − 6.43·23-s − 4.06·25-s + (−3.34 + 5.79i)29-s + (−4.58 + 7.93i)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 + (3.83 + 6.63i)47-s + ⋯
L(s)  = 1  − 0.432·5-s + (−0.421 + 0.906i)7-s − 0.219·11-s + (0.502 + 0.869i)13-s + (−0.848 − 1.47i)17-s + (−0.0798 + 0.138i)19-s − 1.34·23-s − 0.812·25-s + (−0.621 + 1.07i)29-s + (−0.823 + 1.42i)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 + (0.558 + 0.967i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.831 - 0.556i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.831 - 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159506 + 0.525160i\)
\(L(\frac12)\) \(\approx\) \(0.159506 + 0.525160i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 - 2.39i)T \)
good5 \( 1 + 0.967T + 5T^{2} \)
11 \( 1 + 0.728T + 11T^{2} \)
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 \( 1 + 6.43T + 23T^{2} \)
29 \( 1 + (3.34 - 5.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.58 - 7.93i)T + (-15.5 - 26.8i)T^{2} \)
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 \( 1 + (-3.83 - 6.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.05 + 3.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.38 - 4.13i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.46 + 4.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.91 + 5.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.304T + 71T^{2} \)
73 \( 1 + (-5.33 - 9.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.61 - 2.80i)T + (-39.5 + 68.4i)T^{2} \)
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

−10.84736940776482399720386073340, −9.576325403147252671173872778611, −9.109932391903841527265607814369, −8.193360960511074173720831444792, −7.18352174251978402016598321208, −6.33220231146397046727730386346, −5.36111402302508854422272210544, −4.29240854375336502236028378355, −3.16908829752842943690362142423, −1.96438229651644286742686814467, 0.26302009944712920640221313445, 2.11011972250173590065010834329, 3.77117352177935901222975790880, 4.09947854421020449404148254602, 5.71871485126196300071136421239, 6.38137822551560365115889244147, 7.66095552449067476128139903888, 8.014506199654428467560838171541, 9.178298025803645768896914326066, 10.17964100520184135014874549270

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