↑

Properties

Label 2-845-65.9-c1-0-53
Degree $2$
Conductor $845$
Sign $0.998 + 0.0471i$
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  + (0.866 + 0.5i)2-s + (1.73 + i)3-s + (−0.500 − 0.866i)4-s + (2 − i)5-s + (0.999 + 1.73i)6-s − 3i·8-s + (0.499 + 0.866i)9-s + (2.23 + 0.133i)10-s + (1 − 1.73i)11-s − 2i·12-s + (4.46 + 0.267i)15-s + (0.500 − 0.866i)16-s + 0.999i·18-s + (3 + 5.19i)19-s + (−1.86 − 1.23i)20-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.999 + 0.577i)3-s + (−0.250 − 0.433i)4-s + (0.894 − 0.447i)5-s + (0.408 + 0.707i)6-s − 1.06i·8-s + (0.166 + 0.288i)9-s + (0.705 + 0.0423i)10-s + (0.301 − 0.522i)11-s − 0.577i·12-s + (1.15 + 0.0691i)15-s + (0.125 − 0.216i)16-s + 0.235i·18-s + (0.688 + 1.19i)19-s + (−0.417 − 0.275i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.998 + 0.0471i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.998 + 0.0471i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.13741 - 0.0739429i\)
\(L(\frac12)\) \(\approx\) \(3.13741 - 0.0739429i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2 + i)T \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.73 - i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + (-5.19 - 3i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.19 - 3i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.3 + 6i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1 - 1.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.19 - 3i)T + (48.5 - 84.0i)T^{2} \)
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

−9.907328529038814953735807338906, −9.408696745202748156225263422471, −8.694126182902135457108378330119, −7.78115358551412203960746843876, −6.27806895542957722015623287653, −5.87822973795211060609803679622, −4.74208550228399563913522176887, −3.94571516408082612952941248643, −2.88212388264555485261955860788, −1.34035349048800381065460823844, 1.90068555780013447646161977948, 2.62444556002423652189850669084, 3.51417291826586896368346270524, 4.70021068654998012325253263571, 5.72687939116398564035343276555, 6.88676633638802467167309765587, 7.68558540749055724284891714157, 8.469509720473599053871650565389, 9.365445767215822315806664957712, 9.998920395991021469924006860076

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