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Properties

Label 2-2214-369.40-c1-0-40
Degree $2$
Conductor $2214$
Sign $-0.158 + 0.987i$
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 + (1.69 − 2.93i)5-s + (−0.610 + 0.352i)7-s − 0.999·8-s + 3.38·10-s + (−3.96 + 2.28i)11-s + (−0.325 − 0.187i)13-s + (−0.610 − 0.352i)14-s + (−0.5 − 0.866i)16-s − 4.48i·17-s − 6.56i·19-s + (1.69 + 2.93i)20-s + (−3.96 − 2.28i)22-s + (−3.58 + 6.21i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.757 − 1.31i)5-s + (−0.230 + 0.133i)7-s − 0.353·8-s + 1.07·10-s + (−1.19 + 0.690i)11-s + (−0.0902 − 0.0520i)13-s + (−0.163 − 0.0942i)14-s + (−0.125 − 0.216i)16-s − 1.08i·17-s − 1.50i·19-s + (0.378 + 0.656i)20-s + (−0.845 − 0.488i)22-s + (−0.747 + 1.29i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2214\)    =    \(2 \cdot 3^{3} \cdot 41\)
Sign: $-0.158 + 0.987i$
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: \((2,\ 2214,\ (\ :1/2),\ -0.158 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.136656813\)
\(L(\frac12)\) \(\approx\) \(1.136656813\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.740 + 6.36i)T \)
good5 \( 1 + (-1.69 + 2.93i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.610 - 0.352i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.96 - 2.28i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.325 + 0.187i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.48iT - 17T^{2} \)
19 \( 1 + 6.56iT - 19T^{2} \)
23 \( 1 + (3.58 - 6.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.69 - 2.13i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.11 + 5.38i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.18T + 37T^{2} \)
43 \( 1 + (5.57 + 9.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (10.8 - 6.24i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.00iT - 53T^{2} \)
59 \( 1 + (-4.55 + 7.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.65 + 6.32i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.40 + 1.38i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.35iT - 71T^{2} \)
73 \( 1 - 1.83T + 73T^{2} \)
79 \( 1 + (3.73 - 2.15i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.10 + 8.84i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.98iT - 89T^{2} \)
97 \( 1 + (-14.3 + 8.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.876298869070037406626618026808, −7.943067759629149703047507727355, −7.36524369699611667753443733559, −6.38994697167703077015847833797, −5.34651850269571680352303113252, −5.14468099879512930462700542034, −4.30915099534044082181667457599, −2.93003399241576144118948558675, −1.94158394020666461947954900112, −0.31827641032407617708659550998, 1.65330235450201189166051193284, 2.64400121430360481280831608636, 3.24845734190632684950269024616, 4.23323955043150244061882610943, 5.42089416698972585224414840664, 6.19000063762360130579633347459, 6.53626307171906796135926340037, 7.898150319996044549154573849209, 8.340389673653821356298435093703, 9.705102875662349825557564724172

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