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Properties

Label 2-2214-369.40-c1-0-0
Degree $2$
Conductor $2214$
Sign $0.486 + 0.873i$
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 + (−1.84 + 3.18i)5-s + (−3.70 + 2.13i)7-s − 0.999·8-s − 3.68·10-s + (1.47 − 0.852i)11-s + (−2.77 − 1.60i)13-s + (−3.70 − 2.13i)14-s + (−0.5 − 0.866i)16-s + 4.51i·17-s − 5.82i·19-s + (−1.84 − 3.18i)20-s + (1.47 + 0.852i)22-s + (−3.20 + 5.55i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.823 + 1.42i)5-s + (−1.39 + 0.808i)7-s − 0.353·8-s − 1.16·10-s + (0.445 − 0.257i)11-s + (−0.770 − 0.444i)13-s + (−0.989 − 0.571i)14-s + (−0.125 − 0.216i)16-s + 1.09i·17-s − 1.33i·19-s + (−0.411 − 0.713i)20-s + (0.314 + 0.181i)22-s + (−0.669 + 1.15i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2214\)    =    \(2 \cdot 3^{3} \cdot 41\)
Sign: $0.486 + 0.873i$
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: \((2,\ 2214,\ (\ :1/2),\ 0.486 + 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1560839901\)
\(L(\frac12)\) \(\approx\) \(0.1560839901\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.21 - 3.71i)T \)
good5 \( 1 + (1.84 - 3.18i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.70 - 2.13i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.47 + 0.852i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.77 + 1.60i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.51iT - 17T^{2} \)
19 \( 1 + 5.82iT - 19T^{2} \)
23 \( 1 + (3.20 - 5.55i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.74 + 1.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.57 - 4.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.33T + 37T^{2} \)
43 \( 1 + (-4.05 - 7.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.35 - 1.35i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.28iT - 53T^{2} \)
59 \( 1 + (-5.63 + 9.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.09 + 5.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.2 - 6.47i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.62iT - 71T^{2} \)
73 \( 1 + 8.00T + 73T^{2} \)
79 \( 1 + (-7.60 + 4.38i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.74 + 9.95i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.84iT - 89T^{2} \)
97 \( 1 + (-13.7 + 7.93i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.693878443983814089364761388539, −8.825298641873669768490008334144, −7.915724788263281439934963815943, −7.24477910091150452209556503749, −6.40560161981985486722365955776, −6.19950757697261994108622779995, −5.00719244826859911098411928306, −3.77916812873092472828455786841, −3.23152179495535907135850872295, −2.50823623261964309426743743338, 0.05770685686464538089704367105, 0.980707145889274996797869789428, 2.43573427757352556778403978509, 3.73136483596367034974829398883, 4.13048584839508356444762923512, 4.90747459231380922928929871283, 5.91738847333890360819440341402, 6.88877722545411583504106019390, 7.61748592742956549861711253427, 8.576432635886686963546587378405

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