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Properties

Label 2-2214-369.286-c1-0-36
Degree $2$
Conductor $2214$
Sign $-0.999 + 0.0236i$
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 + (1.58 + 2.74i)5-s + (−1.86 − 1.07i)7-s − 0.999·8-s + 3.17·10-s + (−0.206 − 0.119i)11-s + (−6.02 + 3.47i)13-s + (−1.86 + 1.07i)14-s + (−0.5 + 0.866i)16-s − 4.96i·17-s − 6.10i·19-s + (1.58 − 2.74i)20-s + (−0.206 + 0.119i)22-s + (1.71 + 2.97i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.709 + 1.22i)5-s + (−0.704 − 0.406i)7-s − 0.353·8-s + 1.00·10-s + (−0.0623 − 0.0360i)11-s + (−1.67 + 0.964i)13-s + (−0.497 + 0.287i)14-s + (−0.125 + 0.216i)16-s − 1.20i·17-s − 1.40i·19-s + (0.354 − 0.614i)20-s + (−0.0440 + 0.0254i)22-s + (0.358 + 0.620i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2214\)    =    \(2 \cdot 3^{3} \cdot 41\)
Sign: $-0.999 + 0.0236i$
Analytic conductor: \(17.6788\)
Root analytic conductor: \(4.20462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2214} (1639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2214,\ (\ :1/2),\ -0.999 + 0.0236i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4674755078\)
\(L(\frac12)\) \(\approx\) \(0.4674755078\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 - 6.01i)T \)
good5 \( 1 + (-1.58 - 2.74i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.86 + 1.07i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.206 + 0.119i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.02 - 3.47i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.96iT - 17T^{2} \)
19 \( 1 + 6.10iT - 19T^{2} \)
23 \( 1 + (-1.71 - 2.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.10 - 1.79i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.95 + 8.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
43 \( 1 + (1.63 - 2.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.37 + 3.10i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.18iT - 53T^{2} \)
59 \( 1 + (5.69 + 9.85i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.854 + 1.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.46 - 2.57i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.31iT - 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 + (0.375 + 0.216i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.97 + 3.42i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.54iT - 89T^{2} \)
97 \( 1 + (6.84 + 3.95i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.194995640193675412831659044298, −7.60074870429656444532562516351, −6.81528204019060146338379053034, −6.61986286229900466675622876587, −5.28971408804976982502944601992, −4.67902097412789899901655466082, −3.41125776274361291270302059592, −2.73853189804373494667217432833, −2.01827785314635888651991025447, −0.13003272398122776695240352221, 1.59363325289568585969986318829, 2.80305911519185953933521340628, 3.86753423737245628614964651985, 5.00024455223418340911686194861, 5.40083945797328049096901142858, 6.12251951377939532741658449922, 7.02426892719601934858276088299, 7.968500914693274354385742575282, 8.625818615310894156782157404620, 9.254321238209499392477405049282

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