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Properties

Label 2-77-11.4-c1-0-5
Degree $2$
Conductor $77$
Sign $-0.720 + 0.693i$
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  + (−1.60 − 1.16i)2-s + (0.861 − 2.65i)3-s + (0.598 + 1.84i)4-s + (0.0217 − 0.0158i)5-s + (−4.47 + 3.25i)6-s + (0.309 + 0.951i)7-s + (−0.0378 + 0.116i)8-s + (−3.85 − 2.80i)9-s − 0.0534·10-s + (−1.37 − 3.01i)11-s + 5.40·12-s + (3.94 + 2.86i)13-s + (0.613 − 1.88i)14-s + (−0.0231 − 0.0713i)15-s + (3.33 − 2.42i)16-s + (−1.35 + 0.986i)17-s + ⋯
L(s)  = 1  + (−1.13 − 0.824i)2-s + (0.497 − 1.52i)3-s + (0.299 + 0.921i)4-s + (0.00974 − 0.00707i)5-s + (−1.82 + 1.32i)6-s + (0.116 + 0.359i)7-s + (−0.0133 + 0.0411i)8-s + (−1.28 − 0.933i)9-s − 0.0168·10-s + (−0.414 − 0.909i)11-s + 1.55·12-s + (1.09 + 0.795i)13-s + (0.163 − 0.504i)14-s + (−0.00598 − 0.0184i)15-s + (0.833 − 0.605i)16-s + (−0.329 + 0.239i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.720 + 0.693i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ -0.720 + 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.236230 - 0.586468i\)
\(L(\frac12)\) \(\approx\) \(0.236230 - 0.586468i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (1.37 + 3.01i)T \)
good2 \( 1 + (1.60 + 1.16i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.861 + 2.65i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.0217 + 0.0158i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-3.94 - 2.86i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.35 - 0.986i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.424 + 1.30i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.06T + 23T^{2} \)
29 \( 1 + (1.97 + 6.08i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.24 + 2.35i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.161 - 0.495i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.27 - 10.0i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.73T + 43T^{2} \)
47 \( 1 + (2.76 - 8.52i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.20 - 2.32i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.00 - 9.25i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.85 - 4.97i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 2.81T + 67T^{2} \)
71 \( 1 + (1.65 - 1.20i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.23 + 9.95i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.73 + 3.44i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.10 - 1.53i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.21T + 89T^{2} \)
97 \( 1 + (2.74 + 1.99i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.63999577132888721560380970984, −12.99773781473828510529343614099, −11.59534075837537375387850482744, −11.05431645937750535143853500233, −9.206947111930360309630000179416, −8.540348580694560102880670629149, −7.51022562800546449435340663013, −6.03313390460524444412261762693, −2.86717031095119522655279819678, −1.38230635777839221939653233589, 3.61528827076840361165939371500, 5.22511523529738035125100626576, 7.06936057369143443473872140610, 8.377263157286097258405531035802, 9.148374580615971670870091707142, 10.21653881102160540257147714068, 10.81608322035066356384028486564, 12.89968316910431518775548857841, 14.41199989835149934584212050148, 15.37498869530142196368051938464

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