↑

Properties

Label 2-65-65.12-c2-0-3
Degree $2$
Conductor $65$
Sign $-0.649 - 0.760i$
Analytic cond. $1.77112$
Root an. cond. $1.33083$
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  + (−1.78 + 1.78i)2-s + (0.433 − 0.433i)3-s − 2.33i·4-s + (3.33 + 3.72i)5-s + 1.54i·6-s + (−5.63 + 5.63i)7-s + (−2.95 − 2.95i)8-s + 8.62i·9-s + (−12.5 − 0.693i)10-s − 7.95i·11-s + (−1.01 − 1.01i)12-s + (−7.09 + 10.8i)13-s − 20.0i·14-s + (3.06 + 0.169i)15-s + 19.8·16-s + (4.22 + 4.22i)17-s + ⋯
L(s)  = 1  + (−0.890 + 0.890i)2-s + (0.144 − 0.144i)3-s − 0.584i·4-s + (0.667 + 0.744i)5-s + 0.257i·6-s + (−0.805 + 0.805i)7-s + (−0.369 − 0.369i)8-s + 0.958i·9-s + (−1.25 − 0.0693i)10-s − 0.723i·11-s + (−0.0845 − 0.0845i)12-s + (−0.545 + 0.837i)13-s − 1.43i·14-s + (0.204 + 0.0112i)15-s + 1.24·16-s + (0.248 + 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.649 - 0.760i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ -0.649 - 0.760i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.322970 + 0.700349i\)
\(L(\frac12)\) \(\approx\) \(0.322970 + 0.700349i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-3.33 - 3.72i)T \)
13 \( 1 + (7.09 - 10.8i)T \)
good2 \( 1 + (1.78 - 1.78i)T - 4iT^{2} \)
3 \( 1 + (-0.433 + 0.433i)T - 9iT^{2} \)
7 \( 1 + (5.63 - 5.63i)T - 49iT^{2} \)
11 \( 1 + 7.95iT - 121T^{2} \)
17 \( 1 + (-4.22 - 4.22i)T + 289iT^{2} \)
19 \( 1 - 28.4T + 361T^{2} \)
23 \( 1 + (-16.8 + 16.8i)T - 529iT^{2} \)
29 \( 1 + 50.7iT - 841T^{2} \)
31 \( 1 + 5.26iT - 961T^{2} \)
37 \( 1 + (-16.4 + 16.4i)T - 1.36e3iT^{2} \)
41 \( 1 - 61.6iT - 1.68e3T^{2} \)
43 \( 1 + (18.6 - 18.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (-21.8 + 21.8i)T - 2.20e3iT^{2} \)
53 \( 1 + (-38.2 + 38.2i)T - 2.80e3iT^{2} \)
59 \( 1 - 12.8T + 3.48e3T^{2} \)
61 \( 1 + 45.4T + 3.72e3T^{2} \)
67 \( 1 + (-22.7 + 22.7i)T - 4.48e3iT^{2} \)
71 \( 1 + 0.496iT - 5.04e3T^{2} \)
73 \( 1 + (-18.8 - 18.8i)T + 5.32e3iT^{2} \)
79 \( 1 - 73.0iT - 6.24e3T^{2} \)
83 \( 1 + (70.8 + 70.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 2.26T + 7.92e3T^{2} \)
97 \( 1 + (85.2 - 85.2i)T - 9.40e3iT^{2} \)
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

−15.25805659029496641726297302059, −14.13686713135701527591152414926, −13.06779995735238466231189929877, −11.56147835116553424290906888629, −9.996561526649230240278481689546, −9.232236515227932054791473210391, −7.923210975644722882836506905304, −6.74980584329805276645575136639, −5.71765861378959451136331388960, −2.80889000541957575080047390193, 0.984682674648569272687720155218, 3.20359511326260529924792500639, 5.39811867868489540064280515502, 7.24083636581402453656974074484, 9.007279466889804869296688780185, 9.694321724888228832179685694141, 10.38589557782473497432144645657, 12.01927815648541676552570683591, 12.80158610985308308193196636192, 14.10977914353525313778824109671

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