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Properties

Label 2-218-109.63-c1-0-10
Degree $2$
Conductor $218$
Sign $-0.640 + 0.767i$
Analytic cond. $1.74073$
Root an. cond. $1.31937$
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  + 2-s + (−1.64 − 2.85i)3-s + 4-s + (−0.899 − 1.55i)5-s + (−1.64 − 2.85i)6-s + (0.430 + 0.745i)7-s + 8-s + (−3.94 + 6.83i)9-s + (−0.899 − 1.55i)10-s + (0.530 − 0.918i)11-s + (−1.64 − 2.85i)12-s + (−2.94 − 5.09i)13-s + (0.430 + 0.745i)14-s + (−2.96 + 5.14i)15-s + 16-s − 2.26·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.952 − 1.64i)3-s + 0.5·4-s + (−0.402 − 0.697i)5-s + (−0.673 − 1.16i)6-s + (0.162 + 0.281i)7-s + 0.353·8-s + (−1.31 + 2.27i)9-s + (−0.284 − 0.492i)10-s + (0.159 − 0.276i)11-s + (−0.476 − 0.824i)12-s + (−0.816 − 1.41i)13-s + (0.114 + 0.199i)14-s + (−0.766 + 1.32i)15-s + 0.250·16-s − 0.548·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(218\)    =    \(2 \cdot 109\)
Sign: $-0.640 + 0.767i$
Analytic conductor: \(1.74073\)
Root analytic conductor: \(1.31937\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{218} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 218,\ (\ :1/2),\ -0.640 + 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503492 - 1.07615i\)
\(L(\frac12)\) \(\approx\) \(0.503492 - 1.07615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
109 \( 1 + (8.22 + 6.43i)T \)
good3 \( 1 + (1.64 + 2.85i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.899 + 1.55i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.430 - 0.745i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.530 + 0.918i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.94 + 5.09i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.26T + 17T^{2} \)
19 \( 1 - 5.53T + 19T^{2} \)
23 \( 1 - 3.38T + 23T^{2} \)
29 \( 1 + (-4.09 - 7.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.58 + 4.47i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0303 - 0.0525i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.46T + 41T^{2} \)
43 \( 1 - 8.92T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.46 + 6.00i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.77 - 6.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.76 - 6.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.906 - 1.57i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.56T + 71T^{2} \)
73 \( 1 + (5.27 - 9.13i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.74 - 9.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.94 + 12.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.77 - 13.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.90 - 10.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.10818593276491387155997268693, −11.55514909196637492804356513874, −10.46974156514803341180312312989, −8.543530899664277617587046196634, −7.64438007548654130462180485153, −6.79749970888217061656479784721, −5.56523915883364593568233121149, −4.97980632425965205927036407956, −2.73156750137896172017071695132, −0.950511633279247224204518104664, 3.16016899783620534410729567554, 4.34062846544380303426692808610, 4.93923569551330568826536040169, 6.30577571137053563058490356089, 7.21994985056617258590657420771, 9.142114092881072141343783472070, 9.988795137178622294888022364676, 10.92256468311774245253253161763, 11.57772928901081412733577712867, 12.16234208386652142695080212817

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