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Properties

Label 2-756-9.7-c1-0-4
Degree $2$
Conductor $756$
Sign $-0.368 + 0.929i$
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  + (−0.849 − 1.47i)5-s + (0.5 − 0.866i)7-s + (1.23 − 2.14i)11-s + (−0.388 − 0.673i)13-s − 2.81·17-s − 4.98·19-s + (0.356 + 0.616i)23-s + (1.05 − 1.82i)25-s + (2.25 − 3.90i)29-s + (−2.54 − 4.41i)31-s − 1.69·35-s − 6.87·37-s + (−2.93 − 5.08i)41-s + (2.32 − 4.03i)43-s + (6.49 − 11.2i)47-s + ⋯
L(s)  = 1  + (−0.380 − 0.658i)5-s + (0.188 − 0.327i)7-s + (0.373 − 0.646i)11-s + (−0.107 − 0.186i)13-s − 0.681·17-s − 1.14·19-s + (0.0742 + 0.128i)23-s + (0.211 − 0.365i)25-s + (0.418 − 0.725i)29-s + (−0.457 − 0.793i)31-s − 0.287·35-s − 1.13·37-s + (−0.458 − 0.794i)41-s + (0.354 − 0.614i)43-s + (0.947 − 1.64i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.368 + 0.929i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.368 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605662 - 0.892074i\)
\(L(\frac12)\) \(\approx\) \(0.605662 - 0.892074i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.849 + 1.47i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.23 + 2.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.388 + 0.673i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.81T + 17T^{2} \)
19 \( 1 + 4.98T + 19T^{2} \)
23 \( 1 + (-0.356 - 0.616i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.25 + 3.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.54 + 4.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.87T + 37T^{2} \)
41 \( 1 + (2.93 + 5.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.32 + 4.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.49 + 11.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.88T + 53T^{2} \)
59 \( 1 + (-7.14 - 12.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.15 - 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.99 + 6.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 4.98T + 73T^{2} \)
79 \( 1 + (-4.60 + 7.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.40 + 7.63i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.65T + 89T^{2} \)
97 \( 1 + (4.32 - 7.48i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.22935305389894411132051433419, −8.883064979310624538906404120599, −8.609413289445503835948421894116, −7.54110089900494741938049264335, −6.58903832580675710652095447949, −5.57961577256293706558865475213, −4.48400058792800462300849832621, −3.74796477531777792293496291791, −2.18582406938489999961172606820, −0.54107279242424271217566805958, 1.82934031011155115501946505644, 3.05280968488477681841719858188, 4.21620597484777584603073996345, 5.12942821043884064272717730828, 6.50980026653775534434668651504, 6.95252992730951706708446509139, 8.058777565965495798349205800068, 8.898724794941151566235911720486, 9.729993829189870447216341322195, 10.87386068656485914605960198850

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