↑

Properties

Label 2-847-77.76-c1-0-30
Degree $2$
Conductor $847$
Sign $-0.596 - 0.802i$
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  + 2.40i·2-s + 1.72i·3-s − 3.78·4-s − 3.36i·5-s − 4.14·6-s + (2.55 − 0.672i)7-s − 4.28i·8-s + 0.0361·9-s + 8.09·10-s − 6.51i·12-s + 2.55·13-s + (1.61 + 6.15i)14-s + 5.79·15-s + 2.74·16-s + 3.95·17-s + 0.0869i·18-s + ⋯
L(s)  = 1  + 1.70i·2-s + 0.993i·3-s − 1.89·4-s − 1.50i·5-s − 1.69·6-s + (0.967 − 0.254i)7-s − 1.51i·8-s + 0.0120·9-s + 2.56·10-s − 1.88i·12-s + 0.709·13-s + (0.432 + 1.64i)14-s + 1.49·15-s + 0.686·16-s + 0.960·17-s + 0.0204i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.596 - 0.802i$
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: \((2,\ 847,\ (\ :1/2),\ -0.596 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762378 + 1.51627i\)
\(L(\frac12)\) \(\approx\) \(0.762378 + 1.51627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.55 + 0.672i)T \)
11 \( 1 \)
good2 \( 1 - 2.40iT - 2T^{2} \)
3 \( 1 - 1.72iT - 3T^{2} \)
5 \( 1 + 3.36iT - 5T^{2} \)
13 \( 1 - 2.55T + 13T^{2} \)
17 \( 1 - 3.95T + 17T^{2} \)
19 \( 1 - 0.330T + 19T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 + 2.49iT - 29T^{2} \)
31 \( 1 - 0.222iT - 31T^{2} \)
37 \( 1 + 8.38T + 37T^{2} \)
41 \( 1 - 9.53T + 41T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 - 1.23iT - 47T^{2} \)
53 \( 1 - 0.302T + 53T^{2} \)
59 \( 1 - 9.03iT - 59T^{2} \)
61 \( 1 - 5.68T + 61T^{2} \)
67 \( 1 + 6.57T + 67T^{2} \)
71 \( 1 - 5.35T + 71T^{2} \)
73 \( 1 + 6.99T + 73T^{2} \)
79 \( 1 + 2.95iT - 79T^{2} \)
83 \( 1 - 7.89T + 83T^{2} \)
89 \( 1 - 3.66iT - 89T^{2} \)
97 \( 1 + 13.6iT - 97T^{2} \)
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

−10.14105619027954942199561171850, −9.199003690273758281730970179389, −8.724204501956838308998308868627, −7.986504506737026320728981541786, −7.23363099912192804727309779172, −5.84960190948638746065022090909, −5.19661902424817502348141583386, −4.59338439502185421330666869270, −3.88262524903948777136268990318, −1.17006504113172413457713231934, 1.19559173517879817910291304680, 2.11712836659578402016415837035, 3.03227684622373440196368051058, 3.96099306211880853773262919391, 5.32927926260188806000041784936, 6.55874571053542616809332641245, 7.40549377225221244618825233047, 8.250628206710127227500885381383, 9.306574025732667371314164834068, 10.36471199573898407154498671555

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