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Properties

Label 2-218-109.63-c1-0-5
Degree $2$
Conductor $218$
Sign $0.673 - 0.738i$
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 + (1.09 + 1.89i)3-s + 4-s + (0.908 + 1.57i)5-s + (1.09 + 1.89i)6-s + (−2.27 − 3.94i)7-s + 8-s + (−0.906 + 1.56i)9-s + (0.908 + 1.57i)10-s + (−0.369 + 0.640i)11-s + (1.09 + 1.89i)12-s + (0.0935 + 0.162i)13-s + (−2.27 − 3.94i)14-s + (−1.99 + 3.45i)15-s + 16-s − 4.07·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.633 + 1.09i)3-s + 0.5·4-s + (0.406 + 0.703i)5-s + (0.447 + 0.775i)6-s + (−0.861 − 1.49i)7-s + 0.353·8-s + (−0.302 + 0.523i)9-s + (0.287 + 0.497i)10-s + (−0.111 + 0.193i)11-s + (0.316 + 0.548i)12-s + (0.0259 + 0.0449i)13-s + (−0.608 − 1.05i)14-s + (−0.514 + 0.891i)15-s + 0.250·16-s − 0.988·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(218\)    =    \(2 \cdot 109\)
Sign: $0.673 - 0.738i$
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: \((2,\ 218,\ (\ :1/2),\ 0.673 - 0.738i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89558 + 0.836688i\)
\(L(\frac12)\) \(\approx\) \(1.89558 + 0.836688i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-7.93 - 6.78i)T \)
good3 \( 1 + (-1.09 - 1.89i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.908 - 1.57i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.27 + 3.94i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.369 - 0.640i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0935 - 0.162i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.07T + 17T^{2} \)
19 \( 1 + 3.64T + 19T^{2} \)
23 \( 1 + 0.810T + 23T^{2} \)
29 \( 1 + (-4.03 - 6.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.96 + 8.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.869 + 1.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.73T + 41T^{2} \)
43 \( 1 + 4.45T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.34 - 9.25i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.10 + 12.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.823 + 1.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.69 - 11.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.883T + 71T^{2} \)
73 \( 1 + (3.68 - 6.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.92 - 11.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.02 + 3.51i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.65 + 2.87i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.10 + 1.91i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.75791055546609343548374235331, −11.19214949160791710947509725796, −10.24477014799527295497641723692, −10.01182372708461310437156640509, −8.603197246062818478531551143565, −7.05436084847285557327565674050, −6.38854688409050416390024082306, −4.57883867327613859490087526527, −3.82823730323748986348374105032, −2.73161734636423559675795412594, 1.97829936386322038330790299227, 2.96084550691726821334229737799, 4.87687376848636466595131179758, 6.09668854382669004998930917822, 6.80114404959178991579061081304, 8.396233779240529832417938054875, 8.834922378760403326925954174952, 10.18384830702949331837901124339, 11.82987171357991309164239683710, 12.38270446954019321590746394478

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