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Properties

Label 2-2214-369.40-c1-0-13
Degree $2$
Conductor $2214$
Sign $-0.968 - 0.250i$
Analytic cond. $17.6788$
Root an. cond. $4.20462$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.714 + 1.23i)5-s + (−2.06 + 1.19i)7-s − 0.999·8-s − 1.42·10-s + (3.59 − 2.07i)11-s + (3.86 + 2.23i)13-s + (−2.06 − 1.19i)14-s + (−0.5 − 0.866i)16-s + 8.04i·17-s − 0.0146i·19-s + (−0.714 − 1.23i)20-s + (3.59 + 2.07i)22-s + (1.70 − 2.96i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.319 + 0.553i)5-s + (−0.779 + 0.450i)7-s − 0.353·8-s − 0.451·10-s + (1.08 − 0.625i)11-s + (1.07 + 0.618i)13-s + (−0.551 − 0.318i)14-s + (−0.125 − 0.216i)16-s + 1.95i·17-s − 0.00337i·19-s + (−0.159 − 0.276i)20-s + (0.766 + 0.442i)22-s + (0.356 − 0.617i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2214\)    =    \(2 \cdot 3^{3} \cdot 41\)
Sign: $-0.968 - 0.250i$
Analytic conductor: \(17.6788\)
Root analytic conductor: \(4.20462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2214} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2214,\ (\ :1/2),\ -0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.537399351\)
\(L(\frac12)\) \(\approx\) \(1.537399351\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
41 \( 1 + (2.12 - 6.04i)T \)
good5 \( 1 + (0.714 - 1.23i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.06 - 1.19i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.59 + 2.07i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.86 - 2.23i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 8.04iT - 17T^{2} \)
19 \( 1 + 0.0146iT - 19T^{2} \)
23 \( 1 + (-1.70 + 2.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.69 - 2.13i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.83 + 6.63i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.09T + 37T^{2} \)
43 \( 1 + (4.92 + 8.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (11.3 - 6.54i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.16iT - 53T^{2} \)
59 \( 1 + (6.49 - 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.312 + 0.541i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.40 + 3.69i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.1iT - 71T^{2} \)
73 \( 1 + 2.91T + 73T^{2} \)
79 \( 1 + (-10.9 + 6.31i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.00 + 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.614iT - 89T^{2} \)
97 \( 1 + (3.71 - 2.14i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.068266688334117260533874863678, −8.682021940157459658753421730037, −7.84308662602219576439108493206, −6.77463686138776873860530401356, −6.24074643735826130323974344084, −5.90341580606809287569365323984, −4.43721190239621126272740237254, −3.68731208433034811873226901438, −3.10189934214823651407556117050, −1.52899787458215054337745335729, 0.50323940560530691199940109457, 1.53249690061190321614658760297, 3.05038005916126483845294007246, 3.61861610521032542616721465446, 4.60357181914126968060974919101, 5.24398574108749625175931690273, 6.45057051096496074710276467548, 6.93188757330073902480706085407, 8.022487267226276321034185342287, 8.900935023857873544366314985410

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