↑

Properties

Label 2-77-77.6-c1-0-5
Degree $2$
Conductor $77$
Sign $0.755 + 0.655i$
Analytic cond. $0.614848$
Root an. cond. $0.784122$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.478 − 0.155i)2-s + (1.69 − 2.32i)3-s + (−1.41 + 1.02i)4-s + (−0.572 − 0.186i)5-s + (0.447 − 1.37i)6-s + (−1.17 + 2.37i)7-s + (−1.10 + 1.52i)8-s + (−1.63 − 5.03i)9-s − 0.302·10-s + (3.19 + 0.907i)11-s + 5.03i·12-s + (0.723 + 2.22i)13-s + (−0.192 + 1.31i)14-s + (−1.40 + 1.01i)15-s + (0.787 − 2.42i)16-s + (0.353 − 1.08i)17-s + ⋯
L(s)  = 1  + (0.338 − 0.109i)2-s + (0.977 − 1.34i)3-s + (−0.706 + 0.513i)4-s + (−0.256 − 0.0832i)5-s + (0.182 − 0.562i)6-s + (−0.443 + 0.896i)7-s + (−0.391 + 0.538i)8-s + (−0.545 − 1.67i)9-s − 0.0957·10-s + (0.961 + 0.273i)11-s + 1.45i·12-s + (0.200 + 0.617i)13-s + (−0.0514 + 0.351i)14-s + (−0.362 + 0.263i)15-s + (0.196 − 0.605i)16-s + (0.0858 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09468 - 0.408584i\)
\(L(\frac12)\) \(\approx\) \(1.09468 - 0.408584i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.17 - 2.37i)T \)
11 \( 1 + (-3.19 - 0.907i)T \)
good2 \( 1 + (-0.478 + 0.155i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.69 + 2.32i)T + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.572 + 0.186i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (-0.723 - 2.22i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.353 + 1.08i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.10 + 2.98i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.85T + 23T^{2} \)
29 \( 1 + (-1.26 - 1.74i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (5.93 - 1.92i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.99 + 4.35i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-9.51 - 6.91i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + (-5.83 + 8.02i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.89 - 5.82i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.76 - 3.80i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.69 - 8.28i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.46T + 67T^{2} \)
71 \( 1 + (-1.64 + 5.06i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.02 + 1.46i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.13 - 0.367i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.44 + 7.51i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 + (-5.53 + 1.79i)T + (78.4 - 57.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.15392940802271182986944092112, −13.23374726702863252684198654910, −12.41115149964129216401202592882, −11.78031984209473722977876166568, −9.232032647295685038562700112047, −8.727241791774591856047561618712, −7.51850275368157802757350060228, −6.20556356684332081659122213294, −4.01425501463536425259683019886, −2.43121340342579429470280569553, 3.69242774034380369501019128962, 4.20972037923923534009031497880, 5.99097232872223723295466581430, 8.014076889434695446986512810143, 9.219337345911195553579845386352, 9.976509390954055463336845695149, 10.89496330621657914578642414025, 12.84308658397439430541318742288, 13.96084593018325790867992494673, 14.53353829115294696964010095536

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