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Properties

Label 2-77-7.4-c1-0-4
Degree $2$
Conductor $77$
Sign $-0.678 + 0.734i$
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  + (−0.939 − 1.62i)2-s + (−0.326 + 0.565i)3-s + (−0.766 + 1.32i)4-s + (−1.76 − 3.05i)5-s + 1.22·6-s + (0.418 − 2.61i)7-s − 0.879·8-s + (1.28 + 2.22i)9-s + (−3.31 + 5.74i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + 4.41·13-s + (−4.64 + 1.77i)14-s + 2.30·15-s + (2.35 + 4.08i)16-s + (2.62 − 4.54i)17-s + ⋯
L(s)  = 1  + (−0.664 − 1.15i)2-s + (−0.188 + 0.326i)3-s + (−0.383 + 0.663i)4-s + (−0.789 − 1.36i)5-s + 0.500·6-s + (0.158 − 0.987i)7-s − 0.310·8-s + (0.428 + 0.743i)9-s + (−1.04 + 1.81i)10-s + (−0.150 + 0.261i)11-s + (−0.144 − 0.249i)12-s + 1.22·13-s + (−1.24 + 0.473i)14-s + 0.595·15-s + (0.589 + 1.02i)16-s + (0.636 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.678 + 0.734i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ -0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.237537 - 0.542898i\)
\(L(\frac12)\) \(\approx\) \(0.237537 - 0.542898i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.418 + 2.61i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.939 + 1.62i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.326 - 0.565i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.76 + 3.05i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 4.41T + 13T^{2} \)
17 \( 1 + (-2.62 + 4.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.907 + 1.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.16 - 5.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.92T + 29T^{2} \)
31 \( 1 + (0.733 - 1.27i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.22 + 3.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.283T + 41T^{2} \)
43 \( 1 + 3.41T + 43T^{2} \)
47 \( 1 + (-2.27 - 3.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.61 + 6.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.76 - 8.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.573 - 0.994i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.347 + 0.601i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 + (1.17 - 2.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.20 - 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + (-1.73 - 3.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−13.63801196192578073119382962543, −12.76144062044346234471311586770, −11.59186142428630266404540620970, −10.87285674031996839612005480183, −9.759952770622672609531283578259, −8.675838249378897535507185681820, −7.53400516902111194902452304018, −5.07780975599190330713244507662, −3.79479374736086033284608987479, −1.13021797604519816026999231821, 3.38396963030364195163891180328, 6.04866766317231801552341841296, 6.66134489220317241257638117196, 7.918299842298887121279945495677, 8.778969099977385008748664845827, 10.39053295457640230907969598486, 11.59589249101043568080119867579, 12.58401896920947302311404055951, 14.42402216600869875738822759550, 15.19721065691897039893042004406

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