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Properties

Label 2-756-36.23-c1-0-16
Degree $2$
Conductor $756$
Sign $-0.388 + 0.921i$
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  + (−0.498 − 1.32i)2-s + (−1.50 + 1.31i)4-s + (−2.06 − 1.19i)5-s + (0.866 − 0.5i)7-s + (2.49 + 1.33i)8-s + (−0.549 + 3.33i)10-s + (3.21 + 5.57i)11-s + (2.15 − 3.72i)13-s + (−1.09 − 0.896i)14-s + (0.516 − 3.96i)16-s − 1.89i·17-s + 0.529i·19-s + (4.68 − 0.935i)20-s + (5.77 − 7.03i)22-s + (2.64 − 4.58i)23-s + ⋯
L(s)  = 1  + (−0.352 − 0.935i)2-s + (−0.751 + 0.659i)4-s + (−0.925 − 0.534i)5-s + (0.327 − 0.188i)7-s + (0.882 + 0.470i)8-s + (−0.173 + 1.05i)10-s + (0.970 + 1.68i)11-s + (0.596 − 1.03i)13-s + (−0.292 − 0.239i)14-s + (0.129 − 0.991i)16-s − 0.459i·17-s + 0.121i·19-s + (1.04 − 0.209i)20-s + (1.23 − 1.50i)22-s + (0.552 − 0.956i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.388 + 0.921i$
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: \((2,\ 756,\ (\ :1/2),\ -0.388 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561469 - 0.845739i\)
\(L(\frac12)\) \(\approx\) \(0.561469 - 0.845739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.498 + 1.32i)T \)
3 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (2.06 + 1.19i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.21 - 5.57i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.15 + 3.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.89iT - 17T^{2} \)
19 \( 1 - 0.529iT - 19T^{2} \)
23 \( 1 + (-2.64 + 4.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.301 + 0.174i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.09 + 2.94i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.842T + 37T^{2} \)
41 \( 1 + (4.58 + 2.64i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.38 + 4.26i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.04 - 3.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12.5iT - 53T^{2} \)
59 \( 1 + (-2.66 + 4.61i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.49 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.32 - 5.38i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.12T + 71T^{2} \)
73 \( 1 - 5.07T + 73T^{2} \)
79 \( 1 + (-2.87 + 1.66i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.23 + 5.60i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.4iT - 89T^{2} \)
97 \( 1 + (3.86 + 6.69i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.09802040821017203482328517183, −9.305273749327673329928768133757, −8.455851591674769000104872956994, −7.74167646148299214064666759353, −6.89797043897862152607741006866, −5.16513597130658022789481167107, −4.33036003225327699847484843191, −3.61647890571621994548407716351, −2.09539139788206258392989750367, −0.71597099816949439901423762315, 1.25206181704047402618124364237, 3.46317494470370882069613049472, 4.16495363306963121633690424312, 5.54053826920737823257332390839, 6.35546617820043487233718900476, 7.14247096172472239389963096092, 8.034295500979200004370100104465, 8.821787199244529707257701450139, 9.324776714051510642454166495543, 10.84637877738333285092114374244

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