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Properties

Label 2-77-77.76-c1-0-2
Degree $2$
Conductor $77$
Sign $0.882 + 0.470i$
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  − 1.41i·2-s + 2.23i·3-s − 2.23i·5-s + 3.16·6-s + (1.58 + 2.12i)7-s − 2.82i·8-s − 2.00·9-s − 3.16·10-s + (−3 + 1.41i)11-s − 6.32·13-s + (3 − 2.23i)14-s + 5.00·15-s − 4.00·16-s + 2.82i·18-s + 3.16·19-s + ⋯
L(s)  = 1  − 0.999i·2-s + 1.29i·3-s − 0.999i·5-s + 1.29·6-s + (0.597 + 0.801i)7-s − 0.999i·8-s − 0.666·9-s − 1.00·10-s + (−0.904 + 0.426i)11-s − 1.75·13-s + (0.801 − 0.597i)14-s + 1.29·15-s − 1.00·16-s + 0.666i·18-s + 0.725·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.882 + 0.470i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.882 + 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.978841 - 0.244609i\)
\(L(\frac12)\) \(\approx\) \(0.978841 - 0.244609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.58 - 2.12i)T \)
11 \( 1 + (3 - 1.41i)T \)
good2 \( 1 + 1.41iT - 2T^{2} \)
3 \( 1 - 2.23iT - 3T^{2} \)
5 \( 1 + 2.23iT - 5T^{2} \)
13 \( 1 + 6.32T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 3.16T + 19T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 + 6.70iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 - 9.48T + 41T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 - 4.47iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 2.23iT - 59T^{2} \)
61 \( 1 - 3.16T + 61T^{2} \)
67 \( 1 - 11T + 67T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 - 3.16T + 73T^{2} \)
79 \( 1 + 8.48iT - 79T^{2} \)
83 \( 1 + 9.48T + 83T^{2} \)
89 \( 1 + 2.23iT - 89T^{2} \)
97 \( 1 - 6.70iT - 97T^{2} \)
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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

−14.62952926544691300205816133475, −12.83183722072605330609778214000, −12.13155401606035181292274243983, −11.09194271487551417841285780592, −9.864859786780334358578785974913, −9.395875319706378008010643248436, −7.76468001337108502788456218067, −5.29394234540745949716918476449, −4.41761480213862959022213035276, −2.48357509766793217070371542497, 2.47464235229053040124719550329, 5.28719110765115007953962567184, 6.82062635626822745794787763240, 7.34063380060344316724817172191, 8.072187683008054892937878537238, 10.28226613420318595849526213367, 11.36932173789363899884118231400, 12.53509818569497510373937051691, 14.00239147289052012476694589578, 14.30907802043378142686202147827

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