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Properties

Label 2-1338-223.183-c1-0-18
Degree $2$
Conductor $1338$
Sign $0.988 + 0.151i$
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 + (−0.355 + 0.616i)5-s + (−0.5 − 0.866i)6-s − 0.0500·7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.355 + 0.616i)10-s + (−0.119 + 0.206i)11-s + (−0.5 − 0.866i)12-s + 6.08·13-s − 0.0500·14-s + 0.711·15-s + 16-s − 2.89·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.288 − 0.499i)3-s + 0.5·4-s + (−0.159 + 0.275i)5-s + (−0.204 − 0.353i)6-s − 0.0189·7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.112 + 0.194i)10-s + (−0.0358 + 0.0621i)11-s + (−0.144 − 0.249i)12-s + 1.68·13-s − 0.0133·14-s + 0.183·15-s + 0.250·16-s − 0.701·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1338\)    =    \(2 \cdot 3 \cdot 223\)
Sign: $0.988 + 0.151i$
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: \((2,\ 1338,\ (\ :1/2),\ 0.988 + 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.459922287\)
\(L(\frac12)\) \(\approx\) \(2.459922287\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.73 - 14.4i)T \)
good5 \( 1 + (0.355 - 0.616i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.0500T + 7T^{2} \)
11 \( 1 + (0.119 - 0.206i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.08T + 13T^{2} \)
17 \( 1 + 2.89T + 17T^{2} \)
19 \( 1 + (-2.64 - 4.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.63 + 2.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.04 + 7.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.00 - 1.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.18 + 3.78i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + (-0.543 - 0.940i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.00 - 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.23 + 3.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.30T + 59T^{2} \)
61 \( 1 + (-0.798 - 1.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.18 - 3.77i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.74 - 3.02i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.369 - 0.639i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.02 + 6.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.53 - 7.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.61 + 7.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.07 + 1.85i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.714249169910743156673584731020, −8.547259043129217399752629965281, −7.904722156217585312740107062284, −6.93058510293501872433558540335, −6.19755703309963171236170334845, −5.62845265903519806545101498208, −4.39021611680152235681854596558, −3.59351267643750932182123057278, −2.45592472087425252680197533370, −1.17012184107002281221196887989, 1.09048437403689668711629310431, 2.73391958145375538535597132403, 3.72909578923036056265585190402, 4.51736894854699834740590490070, 5.33358483602197536959401447435, 6.23738695836935552903795093680, 6.88601004786326239795464253692, 8.112073147839540289640843075898, 8.821351294608974789214709014152, 9.647091411432925175656547237476

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