Properties

Label 2-4368-13.12-c1-0-8
Degree $2$
Conductor $4368$
Sign $-0.911 - 0.410i$
Analytic cond. $34.8786$
Root an. cond. $5.90581$
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  − 3-s + 4.15i·5-s i·7-s + 9-s − 3.19i·11-s + (1.48 − 3.28i)13-s − 4.15i·15-s + 3.35·17-s + 2.38i·19-s + i·21-s − 0.387·23-s − 12.2·25-s − 27-s − 7.92·29-s + 10.7i·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.85i·5-s − 0.377i·7-s + 0.333·9-s − 0.963i·11-s + (0.410 − 0.911i)13-s − 1.07i·15-s + 0.812·17-s + 0.547i·19-s + 0.218i·21-s − 0.0808·23-s − 2.45·25-s − 0.192·27-s − 1.47·29-s + 1.92i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4368\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.911 - 0.410i$
Analytic conductor: \(34.8786\)
Root analytic conductor: \(5.90581\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4368} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4368,\ (\ :1/2),\ -0.911 - 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7965424219\)
\(L(\frac12)\) \(\approx\) \(0.7965424219\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 + iT \)
13 \( 1 + (-1.48 + 3.28i)T \)
good5 \( 1 - 4.15iT - 5T^{2} \)
11 \( 1 + 3.19iT - 11T^{2} \)
17 \( 1 - 3.35T + 17T^{2} \)
19 \( 1 - 2.38iT - 19T^{2} \)
23 \( 1 + 0.387T + 23T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 - 10.7iT - 31T^{2} \)
37 \( 1 - 1.61iT - 37T^{2} \)
41 \( 1 + 1.45iT - 41T^{2} \)
43 \( 1 + 1.92T + 43T^{2} \)
47 \( 1 + 3.76iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 6.15iT - 59T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 - 5.61iT - 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 - 8.96T + 79T^{2} \)
83 \( 1 - 6.99iT - 83T^{2} \)
89 \( 1 + 0.932iT - 89T^{2} \)
97 \( 1 - 3.35iT - 97T^{2} \)
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

−8.433815198887284284276717049388, −7.86080993691048609198646363466, −7.03325825975905510069053730716, −6.61313957586747623339619870983, −5.72689354483106893558212347362, −5.34625106891075546511936563851, −3.67348197838188564036176441413, −3.54184261611085290789808549306, −2.55519339048367550310426472584, −1.22221734115370871575007070507, 0.25813706167670251128205131677, 1.43533506043107935826415030398, 2.11996741736345741694370347192, 3.79688872754578418594510120229, 4.43636078370550383974812972733, 5.06560451838376738672593092904, 5.70385445627123348248241255594, 6.40911983698721447675713933251, 7.52227046995891792861976157924, 7.977634793706981612085731519719

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