↑

Properties

Label 2-306-17.13-c1-0-5
Degree $2$
Conductor $306$
Sign $0.913 + 0.405i$
Analytic cond. $2.44342$
Root an. cond. $1.56314$
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  + i·2-s − 4-s + (−2 − 2i)5-s + (1 − i)7-s − i·8-s + (2 − 2i)10-s + (2 − 2i)11-s + 6·13-s + (1 + i)14-s + 16-s + (−1 − 4i)17-s − 4i·19-s + (2 + 2i)20-s + (2 + 2i)22-s + (−3 + 3i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.894 − 0.894i)5-s + (0.377 − 0.377i)7-s − 0.353i·8-s + (0.632 − 0.632i)10-s + (0.603 − 0.603i)11-s + 1.66·13-s + (0.267 + 0.267i)14-s + 0.250·16-s + (−0.242 − 0.970i)17-s − 0.917i·19-s + (0.447 + 0.447i)20-s + (0.426 + 0.426i)22-s + (−0.625 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(306\)    =    \(2 \cdot 3^{2} \cdot 17\)
Sign: $0.913 + 0.405i$
Analytic conductor: \(2.44342\)
Root analytic conductor: \(1.56314\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{306} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 306,\ (\ :1/2),\ 0.913 + 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08366 - 0.229803i\)
\(L(\frac12)\) \(\approx\) \(1.08366 - 0.229803i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
17 \( 1 + (1 + 4i)T \)
good5 \( 1 + (2 + 2i)T + 5iT^{2} \)
7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 + (-2 + 2i)T - 11iT^{2} \)
13 \( 1 - 6T + 13T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (3 - 3i)T - 23iT^{2} \)
29 \( 1 + (-2 - 2i)T + 29iT^{2} \)
31 \( 1 + (3 + 3i)T + 31iT^{2} \)
37 \( 1 + (6 + 6i)T + 37iT^{2} \)
41 \( 1 + (1 - i)T - 41iT^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 - 10T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-2 + 2i)T - 61iT^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (-1 - i)T + 71iT^{2} \)
73 \( 1 + (1 + i)T + 73iT^{2} \)
79 \( 1 + (5 - 5i)T - 79iT^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (11 + 11i)T + 97iT^{2} \)
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

−11.53383237076331353896346265809, −10.99204063501587875169903683478, −9.349288390015091112167286375674, −8.656085921386729275390739070308, −7.915371461903695213301966083299, −6.85895005253238785245726957744, −5.67921479878309132841481648321, −4.50790479011145719822752276294, −3.65017507812976024460755029739, −0.919457433327447495734086435265, 1.80396497023660261662572338557, 3.49331243258603852155308560096, 4.15050813176063097862601971021, 5.84082181817088696933743350089, 6.93495799039108633180876062431, 8.205742247527223305275223905593, 8.817969526946263001489372439612, 10.34135022542770929091647528781, 10.78582451079648955639632056849, 11.82699567271427670530295196675

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