↑

Properties

Label 2-845-65.64-c1-0-40
Degree $2$
Conductor $845$
Sign $-0.977 + 0.212i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  − 2.54·2-s − 2.15i·3-s + 4.48·4-s + (−2.08 − 0.817i)5-s + 5.48i·6-s + 2.93·7-s − 6.31·8-s − 1.63·9-s + (5.29 + 2.08i)10-s + 0.635i·11-s − 9.64i·12-s − 7.48·14-s + (−1.76 + 4.48i)15-s + 7.11·16-s − 1.22i·17-s + 4.16·18-s + ⋯
L(s)  = 1  − 1.80·2-s − 1.24i·3-s + 2.24·4-s + (−0.930 − 0.365i)5-s + 2.23i·6-s + 1.11·7-s − 2.23·8-s − 0.545·9-s + (1.67 + 0.658i)10-s + 0.191i·11-s − 2.78i·12-s − 1.99·14-s + (−0.454 + 1.15i)15-s + 1.77·16-s − 0.296i·17-s + 0.981·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.977 + 0.212i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (844, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.977 + 0.212i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0503814 - 0.469704i\)
\(L(\frac12)\) \(\approx\) \(0.0503814 - 0.469704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.08 + 0.817i)T \)
13 \( 1 \)
good2 \( 1 + 2.54T + 2T^{2} \)
3 \( 1 + 2.15iT - 3T^{2} \)
7 \( 1 - 2.93T + 7T^{2} \)
11 \( 1 - 0.635iT - 11T^{2} \)
17 \( 1 + 1.22iT - 17T^{2} \)
19 \( 1 + 1.36iT - 19T^{2} \)
23 \( 1 + 2.15iT - 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 8.96iT - 31T^{2} \)
37 \( 1 + 1.22T + 37T^{2} \)
41 \( 1 + 9.96iT - 41T^{2} \)
43 \( 1 - 1.36iT - 43T^{2} \)
47 \( 1 - 6.16T + 47T^{2} \)
53 \( 1 - 0.642iT - 53T^{2} \)
59 \( 1 - 7.59iT - 59T^{2} \)
61 \( 1 + 2.27T + 61T^{2} \)
67 \( 1 - 8.03T + 67T^{2} \)
71 \( 1 - 2.63iT - 71T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 + 1.03T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + 12.5iT - 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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

−9.571993345994748066576544003950, −8.616513171529395384311602917561, −8.169155296406964755869948786884, −7.31846888431489078634248623331, −7.15635945252784426393279746603, −5.79482604640412423610887555114, −4.32251947865873090880143217871, −2.47104209169413002354414515078, −1.50158100415279447736743198842, −0.45451717468729914537225814703, 1.47279673268487001170429344515, 3.06160085406130358816619584044, 4.16421603923104218837866355345, 5.25769908300448076310404441279, 6.69124233297325153776265662621, 7.59397069154937264689545630208, 8.257808860366297857612992371985, 8.873904825886487731841392493768, 9.779451326528619196153451138563, 10.48977940226546256198789227764

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