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Properties

Label 2-77-7.6-c4-0-21
Degree $2$
Conductor $77$
Sign $0.830 + 0.556i$
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  + 6.07·2-s − 0.250i·3-s + 20.9·4-s − 37.5i·5-s − 1.52i·6-s + (40.7 + 27.2i)7-s + 29.9·8-s + 80.9·9-s − 228. i·10-s − 36.4·11-s − 5.23i·12-s − 190. i·13-s + (247. + 165. i)14-s − 9.40·15-s − 153.·16-s + 358. i·17-s + ⋯
L(s)  = 1  + 1.51·2-s − 0.0277i·3-s + 1.30·4-s − 1.50i·5-s − 0.0422i·6-s + (0.830 + 0.556i)7-s + 0.467·8-s + 0.999·9-s − 2.28i·10-s − 0.301·11-s − 0.0363i·12-s − 1.12i·13-s + (1.26 + 0.845i)14-s − 0.0417·15-s − 0.597·16-s + 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.830 + 0.556i$
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: \((2,\ 77,\ (\ :2),\ 0.830 + 0.556i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.65395 - 1.11092i\)
\(L(\frac12)\) \(\approx\) \(3.65395 - 1.11092i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-40.7 - 27.2i)T \)
11 \( 1 + 36.4T \)
good2 \( 1 - 6.07T + 16T^{2} \)
3 \( 1 + 0.250iT - 81T^{2} \)
5 \( 1 + 37.5iT - 625T^{2} \)
13 \( 1 + 190. iT - 2.85e4T^{2} \)
17 \( 1 - 358. iT - 8.35e4T^{2} \)
19 \( 1 - 439. iT - 1.30e5T^{2} \)
23 \( 1 - 397.T + 2.79e5T^{2} \)
29 \( 1 + 1.57e3T + 7.07e5T^{2} \)
31 \( 1 - 1.02e3iT - 9.23e5T^{2} \)
37 \( 1 - 593.T + 1.87e6T^{2} \)
41 \( 1 + 2.74e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.48e3T + 3.41e6T^{2} \)
47 \( 1 - 2.68e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.34e3T + 7.89e6T^{2} \)
59 \( 1 - 198. iT - 1.21e7T^{2} \)
61 \( 1 - 2.19e3iT - 1.38e7T^{2} \)
67 \( 1 - 1.95e3T + 2.01e7T^{2} \)
71 \( 1 - 814.T + 2.54e7T^{2} \)
73 \( 1 + 7.89e3iT - 2.83e7T^{2} \)
79 \( 1 - 2.23e3T + 3.89e7T^{2} \)
83 \( 1 + 1.17e4iT - 4.74e7T^{2} \)
89 \( 1 + 4.89e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.56e4iT - 8.85e7T^{2} \)
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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

−13.26970494323465681852005030503, −12.70748699609016262649209513586, −12.12217144180724243279159855833, −10.61281064937128015799187780566, −8.925286172697210601030810586158, −7.75606156659921655773010690066, −5.74408165923505513333258300733, −5.03746904622433731100192653567, −3.87189250186034921701307932267, −1.64332839863975181069875383068, 2.39840549465803421578808380899, 3.89030793640759564542951872301, 5.00793375934249896307678653779, 6.77707023039555858860851259311, 7.27706116403237623546990968869, 9.625811186372497482439783164770, 11.18323165382830583864935856051, 11.44525922115231139081128694984, 13.20135407044750131895382563198, 13.78368257734535760548339009208

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