↑

Properties

Label 2-847-77.76-c1-0-13
Degree $2$
Conductor $847$
Sign $-0.688 - 0.725i$
Analytic cond. $6.76332$
Root an. cond. $2.60064$
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  − 0.672i·2-s + 2.65i·3-s + 1.54·4-s + 3.54i·5-s + 1.78·6-s + (−1.10 + 2.40i)7-s − 2.38i·8-s − 4.03·9-s + 2.38·10-s + 4.10i·12-s − 1.10·13-s + (1.61 + 0.742i)14-s − 9.41·15-s + 1.48·16-s + 4.17·17-s + 2.71i·18-s + ⋯
L(s)  = 1  − 0.475i·2-s + 1.53i·3-s + 0.773·4-s + 1.58i·5-s + 0.728·6-s + (−0.416 + 0.908i)7-s − 0.843i·8-s − 1.34·9-s + 0.755·10-s + 1.18i·12-s − 0.305·13-s + (0.432 + 0.198i)14-s − 2.43·15-s + 0.372·16-s + 1.01·17-s + 0.640i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.688 - 0.725i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (846, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.688 - 0.725i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.693162 + 1.61259i\)
\(L(\frac12)\) \(\approx\) \(0.693162 + 1.61259i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.10 - 2.40i)T \)
11 \( 1 \)
good2 \( 1 + 0.672iT - 2T^{2} \)
3 \( 1 - 2.65iT - 3T^{2} \)
5 \( 1 - 3.54iT - 5T^{2} \)
13 \( 1 + 1.10T + 13T^{2} \)
17 \( 1 - 4.17T + 17T^{2} \)
19 \( 1 - 1.24T + 19T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 + 2.04iT - 29T^{2} \)
31 \( 1 + 2.99iT - 31T^{2} \)
37 \( 1 - 2.76T + 37T^{2} \)
41 \( 1 + 5.49T + 41T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 + 6.82iT - 47T^{2} \)
53 \( 1 - 1.07T + 53T^{2} \)
59 \( 1 + 2.28iT - 59T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 - 0.489T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 - 12.4iT - 79T^{2} \)
83 \( 1 - 1.61T + 83T^{2} \)
89 \( 1 - 2.30iT - 89T^{2} \)
97 \( 1 - 12.6iT - 97T^{2} \)
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

−10.45767884009112325146568449568, −9.875741710805956256860265297977, −9.366975206364195034031836822670, −7.964704675853364334904247682956, −6.92726728986398689589117485684, −6.11625825130959145035291597827, −5.21617947000307781732303022384, −3.70410880819395116963389544809, −3.14744969750225067240330088811, −2.37306581843072739914840158256, 0.871873922124263795320492388441, 1.68595494013924636971875541432, 3.16936003017839812973669191367, 4.78981574328594458455202321059, 5.71224874635027473075265908549, 6.57539428830440257679654455061, 7.40005244910073987729486040090, 7.85332843007010694324378608335, 8.683523100427502412476461074301, 9.740033145195673482575723341724

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