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Properties

Label 2-1338-223.183-c1-0-1
Degree $2$
Conductor $1338$
Sign $-0.981 + 0.193i$
Analytic cond. $10.6839$
Root an. cond. $3.26863$
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 + (0.5 + 0.866i)3-s + 4-s + (−0.0934 + 0.161i)5-s + (0.5 + 0.866i)6-s − 4.42·7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.0934 + 0.161i)10-s + (0.307 − 0.531i)11-s + (0.5 + 0.866i)12-s − 5.15·13-s − 4.42·14-s − 0.186·15-s + 16-s − 6.51·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (−0.0418 + 0.0724i)5-s + (0.204 + 0.353i)6-s − 1.67·7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.0295 + 0.0512i)10-s + (0.0925 − 0.160i)11-s + (0.144 + 0.249i)12-s − 1.43·13-s − 1.18·14-s − 0.0482·15-s + 0.250·16-s − 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1338\)    =    \(2 \cdot 3 \cdot 223\)
Sign: $-0.981 + 0.193i$
Analytic conductor: \(10.6839\)
Root analytic conductor: \(3.26863\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1338} (1075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1338,\ (\ :1/2),\ -0.981 + 0.193i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2778182558\)
\(L(\frac12)\) \(\approx\) \(0.2778182558\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.5 - 0.866i)T \)
223 \( 1 + (-8.43 - 12.3i)T \)
good5 \( 1 + (0.0934 - 0.161i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 4.42T + 7T^{2} \)
11 \( 1 + (-0.307 + 0.531i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.15T + 13T^{2} \)
17 \( 1 + 6.51T + 17T^{2} \)
19 \( 1 + (2.91 + 5.04i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.35 - 4.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.692 - 1.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.73 + 4.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.896 + 1.55i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.49T + 41T^{2} \)
43 \( 1 + (-0.554 - 0.960i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.88 - 8.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.75 + 3.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 + (2.76 + 4.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.462 - 0.801i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.828 - 1.43i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.01 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.22 - 9.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.93 - 3.34i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.98 + 6.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.73 - 3.00i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.822584769986825236137394930170, −9.452544280856439442361105454713, −8.634416281833657190084598367344, −7.14729555127576553592705772383, −6.90279380495284044699731551068, −5.83535412946210003458840182029, −4.86759664926780922883266459465, −4.03486847217609642937527097140, −3.04620658519201102405373386510, −2.39072901267242469026693463605, 0.07574317500034148423534403246, 2.15096490556722831106002462169, 2.88922814534718554324420955347, 3.93891882303239385309114384000, 4.85028281487149030832848907399, 6.06987489406801143903519683346, 6.72258874873487566321315031629, 7.16520134795096419972668500958, 8.400795741695932858764280715803, 9.161421786166725287942974328388

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