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Properties

Label 2-77-77.6-c1-0-4
Degree $2$
Conductor $77$
Sign $0.951 + 0.306i$
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  + (1.47 − 0.479i)2-s + (0.332 − 0.241i)4-s + (−1.55 − 2.14i)7-s + (−1.45 + 1.99i)8-s + (0.927 + 2.85i)9-s + (−1.89 − 2.71i)11-s + (−3.32 − 2.41i)14-s + (−1.43 + 4.42i)16-s + (2.73 + 3.76i)18-s + (−4.10 − 3.10i)22-s + 9.58·23-s + (−4.04 − 2.93i)25-s + (−1.03 − 0.335i)28-s + (0.804 + 1.10i)29-s + 2.28i·32-s + ⋯
L(s)  = 1  + (1.04 − 0.339i)2-s + (0.166 − 0.120i)4-s + (−0.587 − 0.809i)7-s + (−0.512 + 0.705i)8-s + (0.309 + 0.951i)9-s + (−0.572 − 0.820i)11-s + (−0.888 − 0.645i)14-s + (−0.359 + 1.10i)16-s + (0.645 + 0.888i)18-s + (−0.875 − 0.662i)22-s + 1.99·23-s + (−0.809 − 0.587i)25-s + (−0.195 − 0.0634i)28-s + (0.149 + 0.205i)29-s + 0.404i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.951 + 0.306i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.951 + 0.306i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32231 - 0.207388i\)
\(L(\frac12)\) \(\approx\) \(1.32231 - 0.207388i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.89 + 2.71i)T \)
good2 \( 1 + (-1.47 + 0.479i)T + (1.61 - 1.17i)T^{2} \)
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 \( 1 - 9.58T + 23T^{2} \)
29 \( 1 + (-0.804 - 1.10i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.10 + 0.803i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.77iT - 43T^{2} \)
47 \( 1 + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.06 - 12.5i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 16.3T + 67T^{2} \)
71 \( 1 + (3.08 - 9.48i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (11.9 - 3.86i)T + (63.9 - 46.4i)T^{2} \)
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

−13.92374870051802676887109206349, −13.44692527275093737141500360077, −12.62468193668392188028796501443, −11.22305887117607108526957313053, −10.36995894797413311362452274922, −8.716293884233968398999267906447, −7.32368017433401186878000202017, −5.66869429989558825096445566019, −4.39772082706543120086919542313, −2.96548008747228776688903917910, 3.20489147426626679176319633097, 4.78608684352023633398155941664, 6.02238261084894950157651000341, 7.11620769384632856283985762010, 9.072355223425545008534339829141, 9.896154080686223219812800428976, 11.72109807027477976765240857780, 12.74594675801112287146221779264, 13.23981664489441197047052066932, 14.87517807739990175651386815912

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