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Properties

Label 2-77-77.10-c1-0-3
Degree $2$
Conductor $77$
Sign $0.993 - 0.111i$
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  + (0.636 − 0.367i)2-s + (1.02 + 0.592i)3-s + (−0.730 + 1.26i)4-s + (0.136 − 0.0789i)5-s + 0.871·6-s + (−1.12 − 2.39i)7-s + 2.54i·8-s + (−0.796 − 1.38i)9-s + (0.0579 − 0.100i)10-s + (−1.84 − 2.75i)11-s + (−1.5 + 0.866i)12-s + 2.14·13-s + (−1.59 − 1.10i)14-s + 0.187·15-s + (−0.527 − 0.912i)16-s + (−2.20 + 3.81i)17-s + ⋯
L(s)  = 1  + (0.449 − 0.259i)2-s + (0.592 + 0.342i)3-s + (−0.365 + 0.632i)4-s + (0.0611 − 0.0352i)5-s + 0.355·6-s + (−0.426 − 0.904i)7-s + 0.898i·8-s + (−0.265 − 0.460i)9-s + (0.0183 − 0.0317i)10-s + (−0.557 − 0.830i)11-s + (−0.433 + 0.249i)12-s + 0.594·13-s + (−0.426 − 0.295i)14-s + 0.0483·15-s + (−0.131 − 0.228i)16-s + (−0.533 + 0.924i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.993 - 0.111i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.993 - 0.111i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18701 + 0.0664611i\)
\(L(\frac12)\) \(\approx\) \(1.18701 + 0.0664611i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.12 + 2.39i)T \)
11 \( 1 + (1.84 + 2.75i)T \)
good2 \( 1 + (-0.636 + 0.367i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.02 - 0.592i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.136 + 0.0789i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 2.14T + 13T^{2} \)
17 \( 1 + (2.20 - 3.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.07 - 5.32i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.20 + 2.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.10iT - 29T^{2} \)
31 \( 1 + (-3.69 - 2.13i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.46 - 5.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.42T + 41T^{2} \)
43 \( 1 + 6.98iT - 43T^{2} \)
47 \( 1 + (-3.21 + 1.85i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.160 + 0.277i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.6 + 6.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.41 - 5.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.32 + 2.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.51T + 71T^{2} \)
73 \( 1 + (-1.76 + 3.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.30 - 0.754i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.49T + 83T^{2} \)
89 \( 1 + (11.7 - 6.78i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.55iT - 97T^{2} \)
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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

−14.10108631859211808006401005527, −13.59873196798691597065015937112, −12.56834387566122359243936528601, −11.29850960183287587164182075269, −10.06024838949814703192377628536, −8.739501075938774568822944322477, −7.85905826268093513981734727137, −5.99566448651308462549900613014, −4.07631561873176543391906371062, −3.22064946749991633992734709173, 2.58868588445288596816947225705, 4.76350958960638271724866005484, 5.97764378189508911444843800850, 7.40591917042747026836656884619, 8.903734850443270327130004516240, 9.753409646146498951010127262544, 11.24503881310762621112255372752, 12.73991707080408310966931923542, 13.52126506508076968231510629760, 14.31043299339491251703180102528

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