Properties

Label 2-845-65.49-c1-0-60
Degree $2$
Conductor $845$
Sign $0.0640 + 0.997i$
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  + (0.165 + 0.286i)2-s + (2.33 − 1.34i)3-s + (0.945 − 1.63i)4-s + (−0.702 − 2.12i)5-s + (0.771 + 0.445i)6-s + (−1.67 + 2.90i)7-s + 1.28·8-s + (2.12 − 3.67i)9-s + (0.492 − 0.552i)10-s + (2.81 − 1.62i)11-s − 5.08i·12-s − 1.10·14-s + (−4.49 − 4.00i)15-s + (−1.67 − 2.90i)16-s + (−1.68 − 0.974i)17-s + 1.40·18-s + ⋯
L(s)  = 1  + (0.116 + 0.202i)2-s + (1.34 − 0.777i)3-s + (0.472 − 0.818i)4-s + (−0.314 − 0.949i)5-s + (0.314 + 0.181i)6-s + (−0.633 + 1.09i)7-s + 0.455·8-s + (0.707 − 1.22i)9-s + (0.155 − 0.174i)10-s + (0.847 − 0.489i)11-s − 1.46i·12-s − 0.296·14-s + (−1.16 − 1.03i)15-s + (−0.419 − 0.726i)16-s + (−0.409 − 0.236i)17-s + 0.331·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0640 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0640 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.0640 + 0.997i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.0640 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92726 - 1.80749i\)
\(L(\frac12)\) \(\approx\) \(1.92726 - 1.80749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.702 + 2.12i)T \)
13 \( 1 \)
good2 \( 1 + (-0.165 - 0.286i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-2.33 + 1.34i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.67 - 2.90i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.81 + 1.62i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.68 + 0.974i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.07 + 0.622i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.33 + 1.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.78iT - 31T^{2} \)
37 \( 1 + (0.974 + 1.68i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.40 + 1.39i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.56 - 4.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.86T + 47T^{2} \)
53 \( 1 - 12.8iT - 53T^{2} \)
59 \( 1 + (-2.19 - 1.26i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.74 + 6.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.00 + 3.47i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.54 + 2.62i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.46T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 - 8.61T + 83T^{2} \)
89 \( 1 + (-8.93 + 5.15i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.63 - 4.56i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.460553617649203756760459516147, −9.074723824349256406963007055376, −8.479700293356076542944223140328, −7.43882673312291907709653675593, −6.57378860967364542885306751759, −5.79114355837739401495448236258, −4.63102829826251062971283640393, −3.24505521615450725112263627158, −2.25307042932772443031060562405, −1.14347771015812256185608192782, 2.18750373548156781306067545325, 3.15446070496381441007489001340, 3.86561739424763708270591902274, 4.28409731981188228251822245384, 6.47122273230921030953036217919, 7.12557634621384339844111533876, 7.82554666095020527481560174567, 8.687608550550029125800690638424, 9.697524911170594142162405835145, 10.26433241357000644717727254333

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