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Properties

Label 2-756-189.5-c1-0-0
Degree $2$
Conductor $756$
Sign $-0.546 - 0.837i$
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  + (−0.230 − 1.71i)3-s + (0.315 − 1.79i)5-s + (0.645 + 2.56i)7-s + (−2.89 + 0.791i)9-s + (−5.18 + 0.913i)11-s + (−4.33 + 5.16i)13-s + (−3.14 − 0.129i)15-s − 5.78·17-s + 1.82i·19-s + (4.25 − 1.70i)21-s + (3.09 − 3.69i)23-s + (1.59 + 0.579i)25-s + (2.02 + 4.78i)27-s + (0.0973 + 0.116i)29-s + (−2.86 − 7.86i)31-s + ⋯
L(s)  = 1  + (−0.133 − 0.991i)3-s + (0.141 − 0.800i)5-s + (0.244 + 0.969i)7-s + (−0.964 + 0.263i)9-s + (−1.56 + 0.275i)11-s + (−1.20 + 1.43i)13-s + (−0.812 − 0.0333i)15-s − 1.40·17-s + 0.419i·19-s + (0.928 − 0.371i)21-s + (0.646 − 0.770i)23-s + (0.318 + 0.115i)25-s + (0.390 + 0.920i)27-s + (0.0180 + 0.0215i)29-s + (−0.514 − 1.41i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.546 - 0.837i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.546 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0688613 + 0.127246i\)
\(L(\frac12)\) \(\approx\) \(0.0688613 + 0.127246i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.230 + 1.71i)T \)
7 \( 1 + (-0.645 - 2.56i)T \)
good5 \( 1 + (-0.315 + 1.79i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (5.18 - 0.913i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (4.33 - 5.16i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + 5.78T + 17T^{2} \)
19 \( 1 - 1.82iT - 19T^{2} \)
23 \( 1 + (-3.09 + 3.69i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.0973 - 0.116i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.86 + 7.86i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (2.28 + 3.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.02 - 3.37i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.904 + 0.329i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (10.7 + 3.89i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-1.23 + 0.711i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.70 + 4.78i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (5.09 - 13.9i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.25 + 7.10i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.35 - 3.08i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.14 - 3.54i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.525 + 2.98i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-3.88 + 3.26i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 8.80T + 89T^{2} \)
97 \( 1 + (0.860 - 2.36i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.85258230225903416223828787828, −9.505841071447587669913934814755, −8.866431532105765032358981715034, −8.072524679468426098355485047279, −7.20700386770107781047402313270, −6.27837945788235020733959835557, −5.19346870507469407857570515110, −4.67186436523991174973011674495, −2.52152738926929170980000507399, −1.98335067893460490924727355526, 0.06718755976898259263187284012, 2.66781308869301425101139656268, 3.32852948596491691133822087918, 4.83483706892751579158261201674, 5.19757517313878107930006571910, 6.58363976364332944910798240472, 7.48993345066657142889895648975, 8.281298560978523248456940988937, 9.454660711987348113538719267123, 10.34424115749810644268672637724

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