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Properties

Label 2-77-7.6-c4-0-16
Degree $2$
Conductor $77$
Sign $0.744 - 0.667i$
Analytic cond. $7.95948$
Root an. cond. $2.82125$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.03·2-s + 3.43i·3-s + 33.4·4-s + 21.5i·5-s + 24.1i·6-s + (−36.4 + 32.7i)7-s + 123.·8-s + 69.1·9-s + 151. i·10-s + 36.4·11-s + 115. i·12-s − 337. i·13-s + (−256. + 230. i)14-s − 73.9·15-s + 330.·16-s − 431. i·17-s + ⋯
L(s)  = 1  + 1.75·2-s + 0.381i·3-s + 2.09·4-s + 0.860i·5-s + 0.671i·6-s + (−0.744 + 0.667i)7-s + 1.92·8-s + 0.854·9-s + 1.51i·10-s + 0.301·11-s + 0.799i·12-s − 1.99i·13-s + (−1.30 + 1.17i)14-s − 0.328·15-s + 1.28·16-s − 1.49i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.744 - 0.667i$
Analytic conductor: \(7.95948\)
Root analytic conductor: \(2.82125\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :2),\ 0.744 - 0.667i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.97560 + 1.52106i\)
\(L(\frac12)\) \(\approx\) \(3.97560 + 1.52106i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (36.4 - 32.7i)T \)
11 \( 1 - 36.4T \)
good2 \( 1 - 7.03T + 16T^{2} \)
3 \( 1 - 3.43iT - 81T^{2} \)
5 \( 1 - 21.5iT - 625T^{2} \)
13 \( 1 + 337. iT - 2.85e4T^{2} \)
17 \( 1 + 431. iT - 8.35e4T^{2} \)
19 \( 1 - 492. iT - 1.30e5T^{2} \)
23 \( 1 + 442.T + 2.79e5T^{2} \)
29 \( 1 - 176.T + 7.07e5T^{2} \)
31 \( 1 + 1.00e3iT - 9.23e5T^{2} \)
37 \( 1 + 2.28e3T + 1.87e6T^{2} \)
41 \( 1 - 883. iT - 2.82e6T^{2} \)
43 \( 1 - 2.96e3T + 3.41e6T^{2} \)
47 \( 1 - 974. iT - 4.87e6T^{2} \)
53 \( 1 + 2.74e3T + 7.89e6T^{2} \)
59 \( 1 + 1.29e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.78e3iT - 1.38e7T^{2} \)
67 \( 1 + 1.33e3T + 2.01e7T^{2} \)
71 \( 1 - 481.T + 2.54e7T^{2} \)
73 \( 1 - 3.53e3iT - 2.83e7T^{2} \)
79 \( 1 - 7.10e3T + 3.89e7T^{2} \)
83 \( 1 - 1.98e3iT - 4.74e7T^{2} \)
89 \( 1 - 6.12e3iT - 6.27e7T^{2} \)
97 \( 1 - 9.99e3iT - 8.85e7T^{2} \)
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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

−13.90282394244026665326938543158, −12.77672882615679985483294867245, −12.13097027948418852005405509755, −10.76412815912246766899483349131, −9.812128120219670313284357253771, −7.53439199387395425254157481650, −6.30201760842400652653597477677, −5.29581404957118902457668054705, −3.69095172761932442443815304444, −2.73642416327581417085178285839, 1.71876957103172448488776282019, 3.87465171223051502982381351832, 4.64982528498242868599586827232, 6.39612723383548752105413346699, 7.05727968288499753209007315218, 9.068943177279491188017889857900, 10.68037174816623227714464111466, 12.09001470889650237305236344964, 12.65804158692650943326148094533, 13.54964974179946623404737133443

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