↑

Properties

Label 2-77-77.76-c1-0-1
Degree $2$
Conductor $77$
Sign $-0.198 - 0.980i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.198 - 0.980i$
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: \((2,\ 77,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.622146 + 0.760924i\)
\(L(\frac12)\) \(\approx\) \(0.622146 + 0.760924i\)
\(L(\frac{3}{2})\) 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 \( 1 - 6.70iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.33201731759558523037100265487, −13.88359473164193211081872424992, −12.97636545107336224872667945002, −11.15772481623848638020650972426, −10.33685579134013357063090394577, −8.951487388338861586861358538532, −8.053007278593077786933402589079, −6.34346893034830953426613045021, −5.15122160219172806297229899427, −3.82069300975700524479502475637, 2.01979387140989027606721966788, 3.25780717647641430075884979789, 6.20543042379025854503750523003, 6.92519303809312994562885024199, 8.414570966927381454188131422040, 10.12998230872479206542644840832, 11.02392676508492289876608055419, 12.10179266529382107865460388769, 12.89598175096312899055499727439, 13.65736267383441107835481381598

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