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Properties

Label 2-77-77.54-c1-0-2
Degree $2$
Conductor $77$
Sign $0.673 - 0.739i$
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  + (1.87 + 1.08i)2-s + (−0.555 + 0.320i)3-s + (1.34 + 2.33i)4-s + (−2.93 − 1.69i)5-s − 1.39·6-s + (2.49 − 0.878i)7-s + 1.51i·8-s + (−1.29 + 2.24i)9-s + (−3.67 − 6.36i)10-s + (−3.01 − 1.37i)11-s + (−1.5 − 0.866i)12-s + 2.36·13-s + (5.63 + 1.05i)14-s + 2.17·15-s + (1.05 − 1.82i)16-s + (1.31 + 2.27i)17-s + ⋯
L(s)  = 1  + (1.32 + 0.766i)2-s + (−0.320 + 0.185i)3-s + (0.674 + 1.16i)4-s + (−1.31 − 0.758i)5-s − 0.567·6-s + (0.943 − 0.332i)7-s + 0.536i·8-s + (−0.431 + 0.747i)9-s + (−1.16 − 2.01i)10-s + (−0.910 − 0.414i)11-s + (−0.433 − 0.249i)12-s + 0.655·13-s + (1.50 + 0.281i)14-s + 0.562·15-s + (0.263 − 0.457i)16-s + (0.318 + 0.551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.673 - 0.739i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.673 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28159 + 0.565991i\)
\(L(\frac12)\) \(\approx\) \(1.28159 + 0.565991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.49 + 0.878i)T \)
11 \( 1 + (3.01 + 1.37i)T \)
good2 \( 1 + (-1.87 - 1.08i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.555 - 0.320i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.93 + 1.69i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 2.36T + 13T^{2} \)
17 \( 1 + (-1.31 - 2.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.70 - 4.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.705 - 1.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.96iT - 29T^{2} \)
31 \( 1 + (2.54 - 1.47i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.699 - 1.21i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.09T + 41T^{2} \)
43 \( 1 - 4.74iT - 43T^{2} \)
47 \( 1 + (4.60 + 2.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.73 - 6.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.610 - 0.352i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.11 + 10.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.98 + 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.81T + 71T^{2} \)
73 \( 1 + (0.618 + 1.07i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.08 - 1.20i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + (2.33 + 1.35i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.92iT - 97T^{2} \)
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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

−14.66492991486530774856073133622, −13.65168696417160442311033467984, −12.65712194965155527580791086415, −11.64005007220479860829675264065, −10.69083471071593035714223341902, −8.160006081745530168162697753175, −7.82929198739877680783339764922, −5.84751129521819174048164302026, −4.82450831265332937774688670081, −3.85392829481407896944005942509, 2.82505853722408420752543010132, 4.19091141006818555497730561055, 5.49498531047969978366069168098, 7.10696430514525892154261782038, 8.492941276868035800165391025401, 10.76474261413311347431900119917, 11.34498075915107098512615281721, 12.05211455939353005575849056280, 13.03829177937034917371166277429, 14.45835940717029372144518919215

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