↑

Properties

Label 2-105-35.19-c2-0-12
Degree $2$
Conductor $105$
Sign $-0.992 + 0.126i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.71 − 0.990i)2-s + (−0.866 − 1.5i)3-s + (−0.0361 − 0.0625i)4-s + (4.45 − 2.26i)5-s + 3.43i·6-s + (−2.11 − 6.67i)7-s + 8.07i·8-s + (−1.5 + 2.59i)9-s + (−9.89 − 0.527i)10-s + (−3.30 − 5.71i)11-s + (−0.0625 + 0.108i)12-s − 21.3·13-s + (−2.98 + 13.5i)14-s + (−7.25 − 4.72i)15-s + (7.85 − 13.6i)16-s + (7.99 + 13.8i)17-s + ⋯
L(s)  = 1  + (−0.858 − 0.495i)2-s + (−0.288 − 0.5i)3-s + (−0.00903 − 0.0156i)4-s + (0.891 − 0.453i)5-s + 0.572i·6-s + (−0.301 − 0.953i)7-s + 1.00i·8-s + (−0.166 + 0.288i)9-s + (−0.989 − 0.0527i)10-s + (−0.300 − 0.519i)11-s + (−0.00521 + 0.00903i)12-s − 1.64·13-s + (−0.213 + 0.967i)14-s + (−0.483 − 0.314i)15-s + (0.490 − 0.850i)16-s + (0.470 + 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.126i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.992 + 0.126i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.992 + 0.126i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0374253 - 0.591053i\)
\(L(\frac12)\) \(\approx\) \(0.0374253 - 0.591053i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 + (-4.45 + 2.26i)T \)
7 \( 1 + (2.11 + 6.67i)T \)
good2 \( 1 + (1.71 + 0.990i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (3.30 + 5.71i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 21.3T + 169T^{2} \)
17 \( 1 + (-7.99 - 13.8i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (17.3 + 10.0i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-1.58 - 0.916i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 4.58T + 841T^{2} \)
31 \( 1 + (-52.5 + 30.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (9.38 + 5.41i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 18.1iT - 1.68e3T^{2} \)
43 \( 1 - 19.8iT - 1.84e3T^{2} \)
47 \( 1 + (-43.8 + 75.9i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-11.5 + 6.69i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (3.31 - 1.91i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-31.1 - 18.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-25.0 + 14.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 98.4T + 5.04e3T^{2} \)
73 \( 1 + (60.2 + 104. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-17.2 + 29.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 70.9T + 6.88e3T^{2} \)
89 \( 1 + (-115. - 66.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 119.T + 9.40e3T^{2} \)
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

−13.02518602604882620103601425625, −11.87281793383343935575003380785, −10.47556857299795396191688892258, −10.05193259674444100532238731833, −8.833072931508330020376909009040, −7.63930928737510741790904491151, −6.17325706668705812097824228994, −4.86756131859284426921938571153, −2.26277807189535689311417236571, −0.58669370075292686485194193657, 2.71854242457030694680159601813, 4.91083896900759124138444267342, 6.26389741614838390132996274144, 7.38907343509565044412454051148, 8.816499295835848922788554329945, 9.777808075071343621055268205502, 10.22770846932065496792795094044, 12.01770303579895812091379996614, 12.80220597185599023714272960529, 14.31193708414962200768631158007

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