↑

Properties

Label 2-3344-3344.1869-c0-0-1
Degree $2$
Conductor $3344$
Sign $0.337 - 0.941i$
Analytic cond. $1.66887$
Root an. cond. $1.29184$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + 1.41·7-s + (0.707 − 0.707i)8-s + (0.866 + 0.5i)9-s + (0.707 + 0.707i)11-s + (0.258 + 0.965i)13-s + (−0.366 + 1.36i)14-s + (0.500 + 0.866i)16-s + (1.22 − 0.707i)17-s + (−0.707 + 0.707i)18-s + (−0.258 − 0.965i)19-s + (−0.866 + 0.500i)22-s + (−0.866 − 0.5i)25-s − 26-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + 1.41·7-s + (0.707 − 0.707i)8-s + (0.866 + 0.5i)9-s + (0.707 + 0.707i)11-s + (0.258 + 0.965i)13-s + (−0.366 + 1.36i)14-s + (0.500 + 0.866i)16-s + (1.22 − 0.707i)17-s + (−0.707 + 0.707i)18-s + (−0.258 − 0.965i)19-s + (−0.866 + 0.500i)22-s + (−0.866 − 0.5i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.337 - 0.941i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (1869, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ 0.337 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.394913253\)
\(L(\frac12)\) \(\approx\) \(1.394913253\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
good3 \( 1 + (-0.866 - 0.5i)T^{2} \)
5 \( 1 + (0.866 + 0.5i)T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
13 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.879846718894510492233025486512, −8.005520073043497583541700561868, −7.44851983848576821776272322385, −6.96596141184051928090804975265, −6.00569926905639076796224153364, −5.07630972975162762460269776599, −4.51261757376037041970930687224, −3.95663775003279547462436106518, −2.07559108785447689006308252709, −1.30423082129111429908694771147, 1.33909534503038004990880925699, 1.58074372687969083174782319261, 3.26114604938030127356557956708, 3.72562512784366484308669326148, 4.64845894879848657029323390424, 5.46738838272568978329602437002, 6.29053941092625945993294034661, 7.78789685402937465263298616851, 7.82810088379861866248827862153, 8.695178738548361305017195482101

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