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Properties

Label 2-77-11.9-c1-0-3
Degree $2$
Conductor $77$
Sign $0.851 - 0.524i$
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  + (−0.435 + 1.34i)2-s + (1.75 − 1.27i)3-s + (0.0112 + 0.00820i)4-s + (−0.565 − 1.74i)5-s + (0.942 + 2.89i)6-s + (−0.809 − 0.587i)7-s + (−2.29 + 1.66i)8-s + (0.519 − 1.59i)9-s + 2.58·10-s + (−2.26 + 2.42i)11-s + 0.0302·12-s + (−1.43 + 4.41i)13-s + (1.14 − 0.828i)14-s + (−3.20 − 2.32i)15-s + (−1.22 − 3.77i)16-s + (−1.69 − 5.20i)17-s + ⋯
L(s)  = 1  + (−0.307 + 0.947i)2-s + (1.01 − 0.734i)3-s + (0.00564 + 0.00410i)4-s + (−0.253 − 0.778i)5-s + (0.384 + 1.18i)6-s + (−0.305 − 0.222i)7-s + (−0.811 + 0.589i)8-s + (0.173 − 0.532i)9-s + 0.816·10-s + (−0.681 + 0.731i)11-s + 0.00872·12-s + (−0.398 + 1.22i)13-s + (0.304 − 0.221i)14-s + (−0.827 − 0.601i)15-s + (−0.306 − 0.944i)16-s + (−0.409 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.851 - 0.524i$
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: \((2,\ 77,\ (\ :1/2),\ 0.851 - 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.986473 + 0.279751i\)
\(L(\frac12)\) \(\approx\) \(0.986473 + 0.279751i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.26 - 2.42i)T \)
good2 \( 1 + (0.435 - 1.34i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.75 + 1.27i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.565 + 1.74i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (1.43 - 4.41i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.69 + 5.20i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.69 + 3.40i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.719T + 23T^{2} \)
29 \( 1 + (-0.948 - 0.689i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.404 - 1.24i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.69 + 1.23i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.741 + 0.538i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.02T + 43T^{2} \)
47 \( 1 + (4.83 - 3.51i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.13 + 9.64i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.21 - 4.51i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.93 + 5.96i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 15.4T + 67T^{2} \)
71 \( 1 + (-4.29 - 13.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.86 + 3.53i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.83 - 14.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.35 + 4.16i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + (-0.745 + 2.29i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.56112350998418511103276441446, −13.70867774524868780255316411417, −12.63581213992444229236177128075, −11.59606216582769172376668235846, −9.453959294125169866932105142160, −8.676786784330314803843832666805, −7.47608507545890421398322183053, −6.97195556981289357971203584607, −4.93724321591215694097329586157, −2.59430838486175112491784265723, 2.78399977587664358326351861803, 3.52475014859992153315838817505, 5.95654267258068700367758432777, 7.82018477655949184263550796693, 9.028879778288673473059081154528, 10.23174270751406879950858093789, 10.64900945503412945779884341193, 12.03395032378759273447415526726, 13.24959021671102118942312534583, 14.65620528860394148643464193143

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