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Properties

Label 2-2214-369.40-c1-0-11
Degree $2$
Conductor $2214$
Sign $-0.890 - 0.454i$
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 + (−0.290 + 0.503i)5-s + (1.28 − 0.743i)7-s − 0.999·8-s − 0.581·10-s + (−1.04 + 0.603i)11-s + (−0.548 − 0.316i)13-s + (1.28 + 0.743i)14-s + (−0.5 − 0.866i)16-s + 4.99i·17-s − 4.61i·19-s + (−0.290 − 0.503i)20-s + (−1.04 − 0.603i)22-s + (−2.97 + 5.14i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.130 + 0.225i)5-s + (0.486 − 0.280i)7-s − 0.353·8-s − 0.183·10-s + (−0.315 + 0.182i)11-s + (−0.152 − 0.0878i)13-s + (0.343 + 0.198i)14-s + (−0.125 − 0.216i)16-s + 1.21i·17-s − 1.05i·19-s + (−0.0650 − 0.112i)20-s + (−0.222 − 0.128i)22-s + (−0.619 + 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2214\)    =    \(2 \cdot 3^{3} \cdot 41\)
Sign: $-0.890 - 0.454i$
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: \((2,\ 2214,\ (\ :1/2),\ -0.890 - 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.426415080\)
\(L(\frac12)\) \(\approx\) \(1.426415080\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.88 - 5.09i)T \)
good5 \( 1 + (0.290 - 0.503i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.28 + 0.743i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.04 - 0.603i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.548 + 0.316i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.99iT - 17T^{2} \)
19 \( 1 + 4.61iT - 19T^{2} \)
23 \( 1 + (2.97 - 5.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.39 + 0.808i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.86 - 3.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.82T + 37T^{2} \)
43 \( 1 + (0.131 + 0.227i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.89 + 2.82i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 + (3.14 - 5.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.48 + 4.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.98 - 1.72i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.02iT - 71T^{2} \)
73 \( 1 + 3.43T + 73T^{2} \)
79 \( 1 + (10.0 - 5.78i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.16 + 2.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.76iT - 89T^{2} \)
97 \( 1 + (-4.40 + 2.54i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.158822220020911195309305752574, −8.499905773863616579227374329897, −7.60270319442973426550443017022, −7.20375129499009892200141054767, −6.22932984111508151005138608682, −5.43213352043473706628929876987, −4.64421838666499035606208640651, −3.80702427253367683622785647885, −2.84278916645281130076064906945, −1.49805553778534927147496432861, 0.43221273844667833349910155975, 1.87012418464654571867933553464, 2.72721437905012140134931649996, 3.80200715911094319625640648163, 4.68513456261413883857674726592, 5.33380604736677649306758584014, 6.20115574644842674360803130141, 7.18543194222906221635921307398, 8.135584493651656567764692693877, 8.705146677272222148152771084187

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