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Properties

Label 2-1338-223.39-c1-0-18
Degree $2$
Conductor $1338$
Sign $0.967 - 0.254i$
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 + (0.652 + 1.13i)5-s + (0.5 − 0.866i)6-s + 3.36·7-s − 8-s + (−0.499 − 0.866i)9-s + (−0.652 − 1.13i)10-s + (−0.271 − 0.470i)11-s + (−0.5 + 0.866i)12-s + 1.71·13-s − 3.36·14-s − 1.30·15-s + 16-s + 1.55·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.288 + 0.499i)3-s + 0.5·4-s + (0.291 + 0.505i)5-s + (0.204 − 0.353i)6-s + 1.27·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.206 − 0.357i)10-s + (−0.0818 − 0.141i)11-s + (−0.144 + 0.249i)12-s + 0.476·13-s − 0.900·14-s − 0.337·15-s + 0.250·16-s + 0.377·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1338\)    =    \(2 \cdot 3 \cdot 223\)
Sign: $0.967 - 0.254i$
Analytic conductor: \(10.6839\)
Root analytic conductor: \(3.26863\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1338} (931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1338,\ (\ :1/2),\ 0.967 - 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.403675143\)
\(L(\frac12)\) \(\approx\) \(1.403675143\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 - 14.7i)T \)
good5 \( 1 + (-0.652 - 1.13i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.36T + 7T^{2} \)
11 \( 1 + (0.271 + 0.470i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.71T + 13T^{2} \)
17 \( 1 - 1.55T + 17T^{2} \)
19 \( 1 + (-3.69 + 6.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.97 + 3.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.60 - 4.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.119 - 0.206i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.09 + 3.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.87T + 41T^{2} \)
43 \( 1 + (-1.01 + 1.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.55 + 6.15i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.96 + 5.13i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.705T + 59T^{2} \)
61 \( 1 + (-1.93 + 3.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.32 - 5.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.295 - 0.512i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.65 - 2.86i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.83 - 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.59 - 9.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.23 + 3.87i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.08 + 3.60i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.700531867112023008665160989830, −8.774882871799731786843060930984, −8.275248937752438397644510426933, −7.18710236624170949178025085765, −6.54965363440761584083844186615, −5.36694892646056871938473495293, −4.77175904445070129557963052011, −3.40133445746474629100040900970, −2.32599853773447087533784270636, −0.954378646663455411376726435111, 1.20482325121514202308814984739, 1.74249903429064436269645741368, 3.28136734281682779364039526363, 4.71953382188583604900190604509, 5.50688491813811592302152763652, 6.27825662818990260182233183718, 7.51724093707576443789547657796, 7.893536651163027986908690094484, 8.663633371665180697485860075422, 9.534890218492509955288834677886

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