Properties

Label 2-77-77.52-c1-0-4
Degree $2$
Conductor $77$
Sign $0.604 + 0.796i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.194 − 0.437i)2-s + (0.465 − 2.18i)3-s + (1.18 + 1.31i)4-s + (−1.33 − 0.140i)5-s + (−0.868 − 0.630i)6-s + (−2.54 + 0.718i)7-s + (1.71 − 0.558i)8-s + (−1.83 − 0.818i)9-s + (−0.322 + 0.558i)10-s + (3.31 + 0.105i)11-s + (3.43 − 1.98i)12-s + (−3.83 + 2.78i)13-s + (−0.181 + 1.25i)14-s + (−0.930 + 2.86i)15-s + (−0.279 + 2.65i)16-s + (1.57 − 0.699i)17-s + ⋯
L(s)  = 1  + (0.137 − 0.309i)2-s + (0.268 − 1.26i)3-s + (0.592 + 0.657i)4-s + (−0.598 − 0.0628i)5-s + (−0.354 − 0.257i)6-s + (−0.962 + 0.271i)7-s + (0.607 − 0.197i)8-s + (−0.612 − 0.272i)9-s + (−0.101 + 0.176i)10-s + (0.999 + 0.0317i)11-s + (0.990 − 0.572i)12-s + (−1.06 + 0.773i)13-s + (−0.0485 + 0.335i)14-s + (−0.240 + 0.739i)15-s + (−0.0698 + 0.664i)16-s + (0.380 − 0.169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.604 + 0.796i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.604 + 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.955365 - 0.474440i\)
\(L(\frac12)\) \(\approx\) \(0.955365 - 0.474440i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.54 - 0.718i)T \)
11 \( 1 + (-3.31 - 0.105i)T \)
good2 \( 1 + (-0.194 + 0.437i)T + (-1.33 - 1.48i)T^{2} \)
3 \( 1 + (-0.465 + 2.18i)T + (-2.74 - 1.22i)T^{2} \)
5 \( 1 + (1.33 + 0.140i)T + (4.89 + 1.03i)T^{2} \)
13 \( 1 + (3.83 - 2.78i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.57 + 0.699i)T + (11.3 - 12.6i)T^{2} \)
19 \( 1 + (1.51 - 1.67i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (0.646 + 1.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.79 + 2.20i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-9.37 + 0.985i)T + (30.3 - 6.44i)T^{2} \)
37 \( 1 + (2.56 - 0.544i)T + (33.8 - 15.0i)T^{2} \)
41 \( 1 + (0.669 + 2.05i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 5.22iT - 43T^{2} \)
47 \( 1 + (4.71 + 4.24i)T + (4.91 + 46.7i)T^{2} \)
53 \( 1 + (1.14 + 10.8i)T + (-51.8 + 11.0i)T^{2} \)
59 \( 1 + (4.36 - 3.92i)T + (6.16 - 58.6i)T^{2} \)
61 \( 1 + (0.621 - 5.91i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (0.828 - 1.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.6 - 9.19i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.78 - 6.42i)T + (-7.63 + 72.6i)T^{2} \)
79 \( 1 + (1.81 - 4.07i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (-6.53 - 4.74i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.96 - 1.70i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.90 - 5.37i)T + (-29.9 + 92.2i)T^{2} \)
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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

−14.03082216896119173638209662398, −12.97527812216058279513087591459, −12.07324205137708138258274629845, −11.76469746464101302778954328559, −9.835977795715361572457753353079, −8.312665242121268357121184317749, −7.22110883449991052781215209450, −6.49794287434830876630095413231, −3.85897764218802391424855310666, −2.23559200050843850416739078718, 3.33369061378338713103072621469, 4.72269524196398180009206525740, 6.28366193785847456584664095463, 7.58955289525907379667118178444, 9.409759725987755894337430115860, 10.07018788554317148601714573163, 11.10934696683432907571872757953, 12.38532602752247205933158664129, 13.99983095729540896294953966012, 15.06345807213881917179718692474

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