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Properties

Label 2-77-77.41-c1-0-4
Degree $2$
Conductor $77$
Sign $-0.376 + 0.926i$
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  + (−1.41 − 1.94i)2-s + (1.63 − 0.531i)3-s + (−1.16 + 3.59i)4-s + (1.97 − 2.72i)5-s + (−3.34 − 2.43i)6-s + (−1.43 + 2.22i)7-s + (4.07 − 1.32i)8-s + (−0.0292 + 0.0212i)9-s − 8.09·10-s + (−3.01 + 1.38i)11-s + 6.51i·12-s + (2.07 − 1.50i)13-s + (6.35 − 0.362i)14-s + (1.79 − 5.51i)15-s + (−2.22 − 1.61i)16-s + (3.20 + 2.32i)17-s + ⋯
L(s)  = 1  + (−0.999 − 1.37i)2-s + (0.945 − 0.307i)3-s + (−0.584 + 1.79i)4-s + (0.885 − 1.21i)5-s + (−1.36 − 0.993i)6-s + (−0.540 + 0.841i)7-s + (1.44 − 0.468i)8-s + (−0.00974 + 0.00708i)9-s − 2.56·10-s + (−0.908 + 0.417i)11-s + 1.88i·12-s + (0.574 − 0.417i)13-s + (1.69 − 0.0969i)14-s + (0.462 − 1.42i)15-s + (−0.555 − 0.403i)16-s + (0.776 + 0.564i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.376 + 0.926i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ -0.376 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.435303 - 0.646510i\)
\(L(\frac12)\) \(\approx\) \(0.435303 - 0.646510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.43 - 2.22i)T \)
11 \( 1 + (3.01 - 1.38i)T \)
good2 \( 1 + (1.41 + 1.94i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (-1.63 + 0.531i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.97 + 2.72i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-2.07 + 1.50i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.20 - 2.32i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.102 + 0.314i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 + (2.36 + 0.770i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.130 - 0.179i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.58 - 7.96i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.94 + 9.06i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 + (-1.17 + 0.383i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.244 - 0.177i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (8.58 + 2.79i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.59 - 3.34i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.57T + 67T^{2} \)
71 \( 1 + (4.33 + 3.14i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.16 + 6.64i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.73 - 2.39i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.38 - 4.64i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 3.66iT - 89T^{2} \)
97 \( 1 + (8.01 + 11.0i)T + (-29.9 + 92.2i)T^{2} \)
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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

−13.49460343011473409176011095365, −12.92006747282450890330222622373, −12.14357391586275279942000690238, −10.52101273811118466620678130252, −9.505631382307554569054523321234, −8.774955856861463260647330190953, −8.038718544755377159272082957958, −5.49304441518004408237974979862, −3.07055011384551684588724811417, −1.80786375459461816454853210122, 3.14660021167636120130438260904, 5.80135386619573678020156827683, 6.86941596105068037309688844933, 7.85313239566063517747117831432, 9.149123703261770752752321850382, 9.940113647729779414697857039562, 10.80672476001523413419203596164, 13.47109862677080483464397087213, 14.14785635989847495977910116342, 14.84049029606263368059430941135

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