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Properties

Label 2-77-11.9-c1-0-5
Degree $2$
Conductor $77$
Sign $-0.256 + 0.966i$
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  + (0.788 − 2.42i)2-s + (0.332 − 0.241i)3-s + (−3.65 − 2.65i)4-s + (1.05 + 3.26i)5-s + (−0.323 − 0.996i)6-s + (−0.809 − 0.587i)7-s + (−5.18 + 3.76i)8-s + (−0.874 + 2.69i)9-s + 8.75·10-s + (−0.0297 − 3.31i)11-s − 1.85·12-s + (−0.672 + 2.06i)13-s + (−2.06 + 1.49i)14-s + (1.13 + 0.827i)15-s + (2.26 + 6.97i)16-s + (−1.39 − 4.29i)17-s + ⋯
L(s)  = 1  + (0.557 − 1.71i)2-s + (0.191 − 0.139i)3-s + (−1.82 − 1.32i)4-s + (0.473 + 1.45i)5-s + (−0.132 − 0.406i)6-s + (−0.305 − 0.222i)7-s + (−1.83 + 1.33i)8-s + (−0.291 + 0.897i)9-s + 2.76·10-s + (−0.00897 − 0.999i)11-s − 0.534·12-s + (−0.186 + 0.573i)13-s + (−0.551 + 0.400i)14-s + (0.294 + 0.213i)15-s + (0.566 + 1.74i)16-s + (−0.338 − 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.256 + 0.966i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ -0.256 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.715349 - 0.929972i\)
\(L(\frac12)\) \(\approx\) \(0.715349 - 0.929972i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (0.0297 + 3.31i)T \)
good2 \( 1 + (-0.788 + 2.42i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.332 + 0.241i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-1.05 - 3.26i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.672 - 2.06i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.39 + 4.29i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.96 + 1.42i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 0.648T + 23T^{2} \)
29 \( 1 + (-1.01 - 0.736i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.48 - 7.64i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.05 + 2.94i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.12 + 1.54i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.46T + 43T^{2} \)
47 \( 1 + (-4.07 + 2.96i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.14 + 12.7i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.20 + 4.50i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.42 - 13.6i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 6.22T + 67T^{2} \)
71 \( 1 + (-1.30 - 4.02i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.79 - 3.48i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.01 + 9.28i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.58 + 7.96i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 4.76T + 89T^{2} \)
97 \( 1 + (2.69 - 8.27i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.95543365404534595710330096619, −13.25328951786051752192671371262, −11.70995569496951734474063846622, −10.94114492581210103570143251575, −10.25500258475070453291082288791, −9.014434015017783468188348873040, −6.97228279745501848203766147613, −5.27486274977739055304281449177, −3.37342464068301906088963377283, −2.39789737135663244476015465875, 4.11910068654873997734361548081, 5.33565273809205721681149831445, 6.30508398134841114817530892232, 7.84213297160630921846679608478, 8.883048206741241640266114841760, 9.663039310984358652618323009622, 12.37718534806673325061756148757, 12.80854413370565544517451810968, 13.90844224797904833140512982090, 15.12128372372740506826941623330

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