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Properties

Label 2-77-77.13-c1-0-0
Degree $2$
Conductor $77$
Sign $0.554 + 0.832i$
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  + (−2.59 − 0.843i)2-s + (4.40 + 3.19i)4-s + (1.55 − 2.14i)7-s + (−5.52 − 7.60i)8-s + (0.927 − 2.85i)9-s + (3.13 + 1.08i)11-s + (−5.83 + 4.24i)14-s + (4.55 + 14.0i)16-s + (−4.81 + 6.62i)18-s + (−7.21 − 5.45i)22-s + 3.36·23-s + (−4.04 + 2.93i)25-s + (13.6 − 4.45i)28-s + (−3.04 + 4.18i)29-s − 21.4i·32-s + ⋯
L(s)  = 1  + (−1.83 − 0.596i)2-s + (2.20 + 1.59i)4-s + (0.587 − 0.809i)7-s + (−1.95 − 2.68i)8-s + (0.309 − 0.951i)9-s + (0.945 + 0.326i)11-s + (−1.56 + 1.13i)14-s + (1.13 + 3.50i)16-s + (−1.13 + 1.56i)18-s + (−1.53 − 1.16i)22-s + 0.700·23-s + (−0.809 + 0.587i)25-s + (2.58 − 0.841i)28-s + (−0.564 + 0.777i)29-s − 3.78i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.554 + 0.832i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.425226 - 0.227748i\)
\(L(\frac12)\) \(\approx\) \(0.425226 - 0.227748i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.55 + 2.14i)T \)
11 \( 1 + (-3.13 - 1.08i)T \)
good2 \( 1 + (2.59 + 0.843i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.36T + 23T^{2} \)
29 \( 1 + (3.04 - 4.18i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.95 + 6.50i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 13.0iT - 43T^{2} \)
47 \( 1 + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.15 - 6.63i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 13.8T + 67T^{2} \)
71 \( 1 + (-0.0272 - 0.0839i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.57 + 0.835i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (78.4 + 57.0i)T^{2} \)
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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

−14.57352531423124203427585643286, −12.73275704813754968382814225044, −11.68767726605641525773112976917, −10.85444620144125860015404165417, −9.701961755082570657302423710435, −8.937913606254475678370381565684, −7.57973266816025702740969860656, −6.71377062309938648770693824469, −3.68223452809524052525377654786, −1.39278900581271508679553308498, 1.94007870270729304110028665313, 5.46482451547966621051434341265, 6.81621386434999215812144164943, 8.048198799348857011334735753900, 8.824045289957561167869707846372, 9.941190405211439348674627653014, 11.05899630457667658923997782251, 11.91495619063024775539342888904, 13.99234998221701628826412520131, 15.15118592944148043678640643998

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