Properties

Label 2-77-7.4-c1-0-5
Degree $2$
Conductor $77$
Sign $-0.449 + 0.893i$
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  + (−0.917 − 1.58i)2-s + (1.09 − 1.90i)3-s + (−0.682 + 1.18i)4-s + (0.317 + 0.550i)5-s − 4.03·6-s + (0.317 + 2.62i)7-s − 1.16·8-s + (−0.917 − 1.58i)9-s + (0.582 − 1.00i)10-s + (0.5 − 0.866i)11-s + (1.50 + 2.59i)12-s − 1.80·13-s + (3.88 − 2.91i)14-s + 1.39·15-s + (2.43 + 4.21i)16-s + (1.41 − 2.45i)17-s + ⋯
L(s)  = 1  + (−0.648 − 1.12i)2-s + (0.634 − 1.09i)3-s + (−0.341 + 0.590i)4-s + (0.142 + 0.246i)5-s − 1.64·6-s + (0.120 + 0.992i)7-s − 0.412·8-s + (−0.305 − 0.529i)9-s + (0.184 − 0.319i)10-s + (0.150 − 0.261i)11-s + (0.433 + 0.750i)12-s − 0.499·13-s + (1.03 − 0.778i)14-s + 0.360·15-s + (0.608 + 1.05i)16-s + (0.343 − 0.595i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.449 + 0.893i$
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: \((2,\ 77,\ (\ :1/2),\ -0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438450 - 0.711756i\)
\(L(\frac12)\) \(\approx\) \(0.438450 - 0.711756i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.317 - 2.62i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.917 + 1.58i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.09 + 1.90i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.317 - 0.550i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.80T + 13T^{2} \)
17 \( 1 + (-1.41 + 2.45i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.78 - 4.81i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.08 + 1.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 + (3.21 - 5.56i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.03 + 5.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.53T + 41T^{2} \)
43 \( 1 + 4.86T + 43T^{2} \)
47 \( 1 + (-1.41 - 2.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.73 - 6.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.90 - 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.16 - 3.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.801 + 1.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.29T + 71T^{2} \)
73 \( 1 + (-7.99 + 13.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.38 + 4.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.23T + 83T^{2} \)
89 \( 1 + (0.182 + 0.315i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.59T + 97T^{2} \)
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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

−14.04052957157739843485601002014, −12.55918558264795160720557839665, −12.15282026018480184890961102392, −10.88909061010808921347956076822, −9.576757156818804024253135938294, −8.645393992514118502655631442366, −7.46548999128718578251289408523, −5.84755281498972572103272440736, −3.00018392625285238886447387149, −1.82956938649959526708922619529, 3.63727125864432032688310576714, 5.18601066721870050876048389686, 6.97300884166097984311781674530, 7.967490264424783029488090657076, 9.288134338527491058066729846521, 9.771689068429543228830677506353, 11.21446965131462365671985978180, 12.98382634995223429557658725931, 14.33975489412468285280248672387, 15.03987601597283250393601055935

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