Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.243 - 0.970i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.167 + 0.167i)2-s + (−0.707 + 0.707i)3-s + 1.94i·4-s + (−2.23 + 0.0836i)5-s − 0.236i·6-s + (−0.0627 + 2.64i)7-s + (−0.658 − 0.658i)8-s − 1.00i·9-s + (0.359 − 0.387i)10-s + 3.98·11-s + (−1.37 − 1.37i)12-s + (−0.500 + 0.500i)13-s + (−0.431 − 0.452i)14-s + (1.52 − 1.63i)15-s − 3.66·16-s + (1.67 + 1.67i)17-s + ⋯
L(s)  = 1  + (−0.118 + 0.118i)2-s + (−0.408 + 0.408i)3-s + 0.972i·4-s + (−0.999 + 0.0373i)5-s − 0.0964i·6-s + (−0.0237 + 0.999i)7-s + (−0.232 − 0.232i)8-s − 0.333i·9-s + (0.113 − 0.122i)10-s + 1.20·11-s + (−0.396 − 0.396i)12-s + (−0.138 + 0.138i)13-s + (−0.115 − 0.120i)14-s + (0.392 − 0.423i)15-s − 0.917·16-s + (0.407 + 0.407i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.243 - 0.970i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1/2),\ -0.243 - 0.970i)\)
\(L(1)\)  \(\approx\)  \(0.451376 + 0.578430i\)
\(L(\frac12)\)  \(\approx\)  \(0.451376 + 0.578430i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (2.23 - 0.0836i)T \)
7 \( 1 + (0.0627 - 2.64i)T \)
good2 \( 1 + (0.167 - 0.167i)T - 2iT^{2} \)
11 \( 1 - 3.98T + 11T^{2} \)
13 \( 1 + (0.500 - 0.500i)T - 13iT^{2} \)
17 \( 1 + (-1.67 - 1.67i)T + 17iT^{2} \)
19 \( 1 - 7.21T + 19T^{2} \)
23 \( 1 + (5.16 + 5.16i)T + 23iT^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 + 4.93iT - 31T^{2} \)
37 \( 1 + (-0.292 + 0.292i)T - 37iT^{2} \)
41 \( 1 - 7.63iT - 41T^{2} \)
43 \( 1 + (-3.65 - 3.65i)T + 43iT^{2} \)
47 \( 1 + (-0.305 - 0.305i)T + 47iT^{2} \)
53 \( 1 + (-5.39 - 5.39i)T + 53iT^{2} \)
59 \( 1 + 6.10T + 59T^{2} \)
61 \( 1 + 7.11iT - 61T^{2} \)
67 \( 1 + (-0.944 + 0.944i)T - 67iT^{2} \)
71 \( 1 - 1.19T + 71T^{2} \)
73 \( 1 + (1.38 - 1.38i)T - 73iT^{2} \)
79 \( 1 + 8.64iT - 79T^{2} \)
83 \( 1 + (-11.9 + 11.9i)T - 83iT^{2} \)
89 \( 1 + 7.82T + 89T^{2} \)
97 \( 1 + (7.43 + 7.43i)T + 97iT^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.26947133843382326589300635404, −12.54424642123591066862001704679, −11.96370538890403351525629517681, −11.33469557304047772781087134837, −9.587028577396067021486546345854, −8.613474094958450424204428127201, −7.54811009310912085852383990632, −6.20902023801625685160530209016, −4.46483236407003359209789044905, −3.26558424233013049874454371669, 1.03403287496494915574867682369, 3.84945293073364477529538313597, 5.34955478134475027726477575338, 6.81180137710398803585408146254, 7.69196529923716764615536167086, 9.331777830964841595344728815374, 10.38127961672762541895265392312, 11.51538537082621851102988593721, 12.03019222179592167598173248662, 13.73045092773093190529177214210

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