Properties

Label 2-1224-1224.707-c0-0-0
Degree $2$
Conductor $1224$
Sign $-0.555 - 0.831i$
Analytic cond. $0.610855$
Root an. cond. $0.781572$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.991 + 0.130i)3-s + (−0.965 + 0.258i)4-s + (−0.258 − 0.965i)6-s + (−0.382 − 0.923i)8-s + (0.965 − 0.258i)9-s + (1.78 + 0.882i)11-s + (0.923 − 0.382i)12-s + (0.866 − 0.5i)16-s + (−0.608 − 0.793i)17-s + (0.382 + 0.923i)18-s + (−0.739 + 1.78i)19-s + (−0.641 + 1.88i)22-s + (0.5 + 0.866i)24-s + (0.793 + 0.608i)25-s + ⋯
L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.991 + 0.130i)3-s + (−0.965 + 0.258i)4-s + (−0.258 − 0.965i)6-s + (−0.382 − 0.923i)8-s + (0.965 − 0.258i)9-s + (1.78 + 0.882i)11-s + (0.923 − 0.382i)12-s + (0.866 − 0.5i)16-s + (−0.608 − 0.793i)17-s + (0.382 + 0.923i)18-s + (−0.739 + 1.78i)19-s + (−0.641 + 1.88i)22-s + (0.5 + 0.866i)24-s + (0.793 + 0.608i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ -0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7595728897\)
\(L(\frac12)\) \(\approx\) \(0.7595728897\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.130 - 0.991i)T \)
3 \( 1 + (0.991 - 0.130i)T \)
17 \( 1 + (0.608 + 0.793i)T \)
good5 \( 1 + (-0.793 - 0.608i)T^{2} \)
7 \( 1 + (0.793 - 0.608i)T^{2} \)
11 \( 1 + (-1.78 - 0.882i)T + (0.608 + 0.793i)T^{2} \)
13 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.739 - 1.78i)T + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.991 + 0.130i)T^{2} \)
29 \( 1 + (0.130 - 0.991i)T^{2} \)
31 \( 1 + (0.608 - 0.793i)T^{2} \)
37 \( 1 + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (0.0983 + 0.0862i)T + (0.130 + 0.991i)T^{2} \)
43 \( 1 + (0.741 - 0.965i)T + (-0.258 - 0.965i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.258 - 0.0340i)T + (0.965 + 0.258i)T^{2} \)
61 \( 1 + (-0.793 + 0.608i)T^{2} \)
67 \( 1 + (-1.37 - 0.793i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.382 + 0.923i)T^{2} \)
73 \( 1 + (-1.75 + 0.349i)T + (0.923 - 0.382i)T^{2} \)
79 \( 1 + (0.608 + 0.793i)T^{2} \)
83 \( 1 + (0.758 - 0.0999i)T + (0.965 - 0.258i)T^{2} \)
89 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
97 \( 1 + (0.991 - 0.869i)T + (0.130 - 0.991i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.885883287156701882071153681679, −9.480974313774381717086626140723, −8.533418554743848193384109679632, −7.44910803127203922689799449509, −6.65930173090793205121409204309, −6.27731833170315496842467057419, −5.18934593478049139825199331900, −4.39497108048035009747711298752, −3.69618442833598835571374720436, −1.45783408385176911054375619124, 0.836786292120324365534751213634, 2.09865433863225657375618903637, 3.58788630847082861228297076850, 4.38257155511252400511555702828, 5.20953900611009756554238900089, 6.40488018392974930046875820294, 6.71378258112847289815422415768, 8.377570821129015667605562038642, 8.964143963478008190147929439452, 9.791389052749311862207033069264

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