Properties

Label 2-77-77.25-c1-0-5
Degree $2$
Conductor $77$
Sign $-0.0694 + 0.997i$
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  + (−0.564 − 0.251i)2-s + (−0.978 − 0.207i)3-s + (−1.08 − 1.20i)4-s + (0.233 − 2.22i)5-s + (0.5 + 0.363i)6-s + (1.59 − 2.11i)7-s + (0.690 + 2.12i)8-s + (−1.82 − 0.813i)9-s + (−0.690 + 1.19i)10-s + (2.04 + 2.60i)11-s + (0.809 + 1.40i)12-s + (−0.690 + 0.502i)13-s + (−1.43 + 0.790i)14-s + (−0.690 + 2.12i)15-s + (−0.193 + 1.84i)16-s + (−1.12 + 0.502i)17-s + ⋯
L(s)  = 1  + (−0.399 − 0.177i)2-s + (−0.564 − 0.120i)3-s + (−0.541 − 0.601i)4-s + (0.104 − 0.994i)5-s + (0.204 + 0.148i)6-s + (0.603 − 0.797i)7-s + (0.244 + 0.751i)8-s + (−0.609 − 0.271i)9-s + (−0.218 + 0.378i)10-s + (0.617 + 0.786i)11-s + (0.233 + 0.404i)12-s + (−0.191 + 0.139i)13-s + (−0.382 + 0.211i)14-s + (−0.178 + 0.549i)15-s + (−0.0484 + 0.460i)16-s + (−0.273 + 0.121i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0694 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.0694 + 0.997i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ -0.0694 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.413630 - 0.443419i\)
\(L(\frac12)\) \(\approx\) \(0.413630 - 0.443419i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.59 + 2.11i)T \)
11 \( 1 + (-2.04 - 2.60i)T \)
good2 \( 1 + (0.564 + 0.251i)T + (1.33 + 1.48i)T^{2} \)
3 \( 1 + (0.978 + 0.207i)T + (2.74 + 1.22i)T^{2} \)
5 \( 1 + (-0.233 + 2.22i)T + (-4.89 - 1.03i)T^{2} \)
13 \( 1 + (0.690 - 0.502i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.12 - 0.502i)T + (11.3 - 12.6i)T^{2} \)
19 \( 1 + (-5.25 + 5.83i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (-3.11 - 5.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.354 - 1.08i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.691 + 6.58i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (4.97 - 1.05i)T + (33.8 - 15.0i)T^{2} \)
41 \( 1 + (1.42 + 4.39i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.85T + 43T^{2} \)
47 \( 1 + (5.66 - 6.29i)T + (-4.91 - 46.7i)T^{2} \)
53 \( 1 + (-0.532 - 5.06i)T + (-51.8 + 11.0i)T^{2} \)
59 \( 1 + (-2.48 - 2.75i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (0.725 - 6.90i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (5.30 - 9.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.23 + 3.07i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.77 - 3.08i)T + (-7.63 + 72.6i)T^{2} \)
79 \( 1 + (-0.646 - 0.288i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (-7.28 - 5.29i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (5.23 + 9.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.30 - 1.67i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.09932087117300757268139503864, −13.18629771784422113562914217308, −11.79956167175412017035060281797, −10.98191828245050757410511598021, −9.552545885528550834848879795168, −8.845094165083717309390267109446, −7.24913816200202102064102658967, −5.46576790372544036688142814244, −4.55355169743199541460354082713, −1.12364637577154910260252882813, 3.18355983905936257973913749824, 5.16924825591601117619993026763, 6.54569970800395499160474917463, 8.027094011409560594288940648173, 8.976856340016117597389206383568, 10.41974807761003958908077253016, 11.47587315297298888115241319186, 12.35038090193135632408460533683, 13.98134496068663288278368025031, 14.57433483970111885202968491573

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