Properties

Label 2-756-28.19-c1-0-1
Degree $2$
Conductor $756$
Sign $-0.849 - 0.527i$
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  + (−1.35 − 0.402i)2-s + (1.67 + 1.09i)4-s + (2.11 + 1.22i)5-s + (−1.64 − 2.06i)7-s + (−1.83 − 2.15i)8-s + (−2.38 − 2.51i)10-s + (−5.38 + 3.11i)11-s − 2.57i·13-s + (1.40 + 3.46i)14-s + (1.61 + 3.65i)16-s + (−5.44 + 3.14i)17-s + (−2.28 + 3.96i)19-s + (2.21 + 4.36i)20-s + (8.55 − 2.04i)22-s + (1.66 + 0.963i)23-s + ⋯
L(s)  = 1  + (−0.958 − 0.284i)2-s + (0.837 + 0.545i)4-s + (0.947 + 0.547i)5-s + (−0.622 − 0.782i)7-s + (−0.647 − 0.761i)8-s + (−0.752 − 0.794i)10-s + (−1.62 + 0.938i)11-s − 0.713i·13-s + (0.374 + 0.927i)14-s + (0.403 + 0.914i)16-s + (−1.32 + 0.762i)17-s + (−0.525 + 0.909i)19-s + (0.495 + 0.976i)20-s + (1.82 − 0.436i)22-s + (0.348 + 0.200i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.849 - 0.527i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.849 - 0.527i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0620794 + 0.217765i\)
\(L(\frac12)\) \(\approx\) \(0.0620794 + 0.217765i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.402i)T \)
3 \( 1 \)
7 \( 1 + (1.64 + 2.06i)T \)
good5 \( 1 + (-2.11 - 1.22i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.38 - 3.11i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.57iT - 13T^{2} \)
17 \( 1 + (5.44 - 3.14i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.28 - 3.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.66 - 0.963i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.67T + 29T^{2} \)
31 \( 1 + (1.50 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.60 + 2.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.33iT - 41T^{2} \)
43 \( 1 - 5.77iT - 43T^{2} \)
47 \( 1 + (-4.53 + 7.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.91 - 5.04i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.10 - 1.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.57 + 4.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.4 - 7.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.552iT - 71T^{2} \)
73 \( 1 + (-6.82 + 3.93i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.7 - 6.20i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.324T + 83T^{2} \)
89 \( 1 + (8.69 + 5.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.2iT - 97T^{2} \)
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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

−10.61195804622347654038487205117, −9.975568434357339266144597033235, −9.268159246447145550727918124881, −8.036123449624297918229176502079, −7.40084196003247356280669024707, −6.49454662084585765719575395300, −5.66221136885089023450177431530, −4.05782157583201904721466017520, −2.75886368362648249849160450133, −1.92834930406166167515144880423, 0.13808188888289643512169260785, 2.07573887866032698816087436770, 2.82553610738029520403981338487, 4.96862261386493585638117733973, 5.68508287933274558368183490374, 6.48057275171162259143204754905, 7.43439629987996425414214260806, 8.727965572192188619589673676544, 8.974524788077047645082433462323, 9.721410058141872610329687792760

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