↑

Properties

Label 2-135-15.8-c1-0-0
Degree $2$
Conductor $135$
Sign $0.340 - 0.940i$
Analytic cond. $1.07798$
Root an. cond. $1.03825$
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  + (−1.54 − 1.54i)2-s + 2.79i·4-s + (−1.22 + 1.87i)5-s + (−2.79 + 2.79i)7-s + (1.22 − 1.22i)8-s + (4.79 − 1.00i)10-s + 1.80i·11-s + (−2.79 − 2.79i)13-s + 8.64·14-s + 1.79·16-s + (1.87 + 1.87i)17-s + 3i·19-s + (−5.22 − 3.41i)20-s + (2.79 − 2.79i)22-s + (0.578 − 0.578i)23-s + ⋯
L(s)  = 1  + (−1.09 − 1.09i)2-s + 1.39i·4-s + (−0.547 + 0.836i)5-s + (−1.05 + 1.05i)7-s + (0.433 − 0.433i)8-s + (1.51 − 0.316i)10-s + 0.543i·11-s + (−0.774 − 0.774i)13-s + 2.30·14-s + 0.447·16-s + (0.453 + 0.453i)17-s + 0.688i·19-s + (−1.16 − 0.764i)20-s + (0.595 − 0.595i)22-s + (0.120 − 0.120i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.340 - 0.940i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1/2),\ 0.340 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259304 + 0.181808i\)
\(L(\frac12)\) \(\approx\) \(0.259304 + 0.181808i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.22 - 1.87i)T \)
good2 \( 1 + (1.54 + 1.54i)T + 2iT^{2} \)
7 \( 1 + (2.79 - 2.79i)T - 7iT^{2} \)
11 \( 1 - 1.80iT - 11T^{2} \)
13 \( 1 + (2.79 + 2.79i)T + 13iT^{2} \)
17 \( 1 + (-1.87 - 1.87i)T + 17iT^{2} \)
19 \( 1 - 3iT - 19T^{2} \)
23 \( 1 + (-0.578 + 0.578i)T - 23iT^{2} \)
29 \( 1 + 6.83T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + (5 - 5i)T - 37iT^{2} \)
41 \( 1 - 1.80iT - 41T^{2} \)
43 \( 1 + (-7.79 - 7.79i)T + 43iT^{2} \)
47 \( 1 + (-3.09 - 3.09i)T + 47iT^{2} \)
53 \( 1 + (-7.41 + 7.41i)T - 53iT^{2} \)
59 \( 1 - 6.83T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + (0.582 - 0.582i)T - 67iT^{2} \)
71 \( 1 - 1.80iT - 71T^{2} \)
73 \( 1 + (-3.37 - 3.37i)T + 73iT^{2} \)
79 \( 1 + 1.41iT - 79T^{2} \)
83 \( 1 + (6.25 - 6.25i)T - 83iT^{2} \)
89 \( 1 + 5.41T + 89T^{2} \)
97 \( 1 + (-5.58 + 5.58i)T - 97iT^{2} \)
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

−12.73889271952246193103508057796, −12.27271091170012905014773756163, −11.27660891919444376419692534938, −10.16491882722632224834844236682, −9.650900009156398241954728117416, −8.410014901770799014628517923782, −7.33937081309159652060848978469, −5.82352491071541071157787416520, −3.46212704394241727777177098597, −2.44134445619552269275575750105, 0.44622292915699067578095683676, 3.84325090656324991991826574244, 5.52860323924521259818413439217, 7.02261478097559972252991898916, 7.48546583769218049724673388103, 8.906305007204861449505464615264, 9.457912448851556051245140126826, 10.61234424837008925792413450667, 12.03898281162709884643253559616, 13.14823607935961703029033993874

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