↑

Properties

Label 2-845-13.3-c1-0-49
Degree $2$
Conductor $845$
Sign $-0.859 - 0.511i$
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  + (−0.309 − 0.535i)2-s + (−1.11 − 1.93i)3-s + (0.809 − 1.40i)4-s − 5-s + (−0.690 + 1.19i)6-s + (2.11 − 3.66i)7-s − 2.23·8-s + (−1 + 1.73i)9-s + (0.309 + 0.535i)10-s + (−0.118 − 0.204i)11-s − 3.61·12-s − 2.61·14-s + (1.11 + 1.93i)15-s + (−0.927 − 1.60i)16-s + (1.73 − 3.00i)17-s + 1.23·18-s + ⋯
L(s)  = 1  + (−0.218 − 0.378i)2-s + (−0.645 − 1.11i)3-s + (0.404 − 0.700i)4-s − 0.447·5-s + (−0.282 + 0.488i)6-s + (0.800 − 1.38i)7-s − 0.790·8-s + (−0.333 + 0.577i)9-s + (0.0977 + 0.169i)10-s + (−0.0355 − 0.0616i)11-s − 1.04·12-s − 0.699·14-s + (0.288 + 0.500i)15-s + (−0.231 − 0.401i)16-s + (0.421 − 0.729i)17-s + 0.291·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.859 - 0.511i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.287820 + 1.04725i\)
\(L(\frac12)\) \(\approx\) \(0.287820 + 1.04725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
13 \( 1 \)
good2 \( 1 + (0.309 + 0.535i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.11 + 1.93i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.11 + 3.66i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.118 + 0.204i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.73 + 3.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.11 + 3.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.88 + 3.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.73 - 6.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.97 - 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.11 + 5.40i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.94T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (0.354 - 0.613i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.20 - 12.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.35 + 2.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.11 - 5.40i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.73 - 3.00i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.01933799698946758803849633744, −8.837903761111679287074311155867, −7.62189278834995121858423062145, −7.18580496167336078645866887550, −6.43197583303019262568765205480, −5.34476106833979977375902615982, −4.36633603800195727529310355161, −2.79672712140483236028118505322, −1.31506866949456603414308599452, −0.68669332897087479162575002553, 2.21081435980133761275670491911, 3.55163090941321183731862603101, 4.43567419913265968778768712635, 5.61186309718174485198965012310, 6.00784879807341044619043956536, 7.54993299220743744092741485971, 8.107013760266529085343730976508, 8.952768489495651845545116211183, 9.776733173857354274781714015356, 10.81308303907835108765258175071

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