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Properties

Label 2-77-11.3-c1-0-4
Degree $2$
Conductor $77$
Sign $-0.358 + 0.933i$
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  + (−0.901 + 0.655i)2-s + (−0.883 − 2.71i)3-s + (−0.234 + 0.720i)4-s + (−2.79 − 2.03i)5-s + (2.57 + 1.87i)6-s + (0.309 − 0.951i)7-s + (−0.949 − 2.92i)8-s + (−4.18 + 3.04i)9-s + 3.85·10-s + (3.31 + 0.0938i)11-s + 2.16·12-s + (1.66 − 1.21i)13-s + (0.344 + 1.05i)14-s + (−3.05 + 9.39i)15-s + (1.54 + 1.12i)16-s + (−1.56 − 1.13i)17-s + ⋯
L(s)  = 1  + (−0.637 + 0.463i)2-s + (−0.510 − 1.56i)3-s + (−0.117 + 0.360i)4-s + (−1.25 − 0.908i)5-s + (1.05 + 0.764i)6-s + (0.116 − 0.359i)7-s + (−0.335 − 1.03i)8-s + (−1.39 + 1.01i)9-s + 1.21·10-s + (0.999 + 0.0283i)11-s + 0.625·12-s + (0.462 − 0.335i)13-s + (0.0920 + 0.283i)14-s + (−0.788 + 2.42i)15-s + (0.386 + 0.280i)16-s + (−0.379 − 0.275i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.358 + 0.933i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ -0.358 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.238054 - 0.346270i\)
\(L(\frac12)\) \(\approx\) \(0.238054 - 0.346270i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-3.31 - 0.0938i)T \)
good2 \( 1 + (0.901 - 0.655i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.883 + 2.71i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.79 + 2.03i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1.66 + 1.21i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.56 + 1.13i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.501 - 1.54i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.807T + 23T^{2} \)
29 \( 1 + (-2.46 + 7.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.637 - 0.463i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.10 + 9.56i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.657 + 2.02i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.08T + 43T^{2} \)
47 \( 1 + (-2.33 - 7.19i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.75 - 6.36i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.01 - 3.13i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.871 - 0.632i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.40T + 67T^{2} \)
71 \( 1 + (-2.57 - 1.87i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.378 + 1.16i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.67 + 5.57i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-13.0 - 9.44i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 4.43T + 89T^{2} \)
97 \( 1 + (5.23 - 3.80i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.87386521536583530557639028546, −12.74653684214313439346246948129, −12.18721312245203477926076511245, −11.30092341230755457450175257677, −9.091096345077925969385782433464, −8.014593095135312942032151112706, −7.47236087200382735847346612896, −6.26109312490533133273424012004, −4.09718573179734231953423801607, −0.75956679774041450742393153840, 3.49036956941392011556591088240, 4.78739922937719734920564104709, 6.44454525568981877549202676158, 8.478992235353505226873407750052, 9.444589388585697563226796665217, 10.55461856290786464161793317974, 11.25621677964043527419582775240, 11.81199278298707664071522558902, 14.31678327579370015184437224040, 15.01956095650010782607057151528

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