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Properties

Label 2-756-7.4-c1-0-8
Degree $2$
Conductor $756$
Sign $0.187 + 0.982i$
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  + (−1.21 − 2.09i)5-s + (1.85 + 1.88i)7-s + (2.35 − 4.07i)11-s − 3.42·13-s + (0.851 − 1.47i)17-s + (−0.641 − 1.11i)19-s + (−0.562 − 0.974i)23-s + (−0.430 + 0.746i)25-s + 4.70·29-s + (1.71 − 2.96i)31-s + (1.72 − 6.17i)35-s + (−4.27 − 7.40i)37-s − 3.71·41-s + 5.54·43-s + (−5.91 − 10.2i)47-s + ⋯
L(s)  = 1  + (−0.541 − 0.937i)5-s + (0.699 + 0.714i)7-s + (0.709 − 1.22i)11-s − 0.948·13-s + (0.206 − 0.357i)17-s + (−0.147 − 0.254i)19-s + (−0.117 − 0.203i)23-s + (−0.0861 + 0.149i)25-s + 0.873·29-s + (0.307 − 0.532i)31-s + (0.290 − 1.04i)35-s + (−0.702 − 1.21i)37-s − 0.580·41-s + 0.845·43-s + (−0.862 − 1.49i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.187 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03184 - 0.853760i\)
\(L(\frac12)\) \(\approx\) \(1.03184 - 0.853760i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.85 - 1.88i)T \)
good5 \( 1 + (1.21 + 2.09i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.35 + 4.07i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.42T + 13T^{2} \)
17 \( 1 + (-0.851 + 1.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.641 + 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.562 + 0.974i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.70T + 29T^{2} \)
31 \( 1 + (-1.71 + 2.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.27 + 7.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.71T + 41T^{2} \)
43 \( 1 - 5.54T + 43T^{2} \)
47 \( 1 + (5.91 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.13 + 8.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.06 + 3.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.62 - 8.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.56 + 9.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + (1.06 - 1.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.26 - 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.42T + 83T^{2} \)
89 \( 1 + (-8.04 - 13.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−10.10320734175600489400191636719, −8.991833618436407203585080389843, −8.576920081656051125607729342771, −7.80032455774992161500963293398, −6.63708370166040662870312000845, −5.48306852314886729840614944427, −4.81744284463368629970372125983, −3.73881449858023001642753825111, −2.32976847845232707851885135218, −0.72438325538532723074550833489, 1.60217678453446610127157597266, 3.01680077458057651211961222483, 4.18770447212822073340441491606, 4.87822381829210239425774100799, 6.39899195473838974325474853011, 7.21916780906514629898219558293, 7.65730443052436874405863546215, 8.763357715636282587376731565316, 10.04354052538162876232332127744, 10.33320278997075088377117813570

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