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Properties

Label 2-1338-223.39-c1-0-30
Degree $2$
Conductor $1338$
Sign $-0.997 + 0.0712i$
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 + (−1.93 − 3.35i)5-s + (0.5 − 0.866i)6-s − 0.151·7-s − 8-s + (−0.499 − 0.866i)9-s + (1.93 + 3.35i)10-s + (0.443 + 0.768i)11-s + (−0.5 + 0.866i)12-s − 2.55·13-s + 0.151·14-s + 3.87·15-s + 16-s + 6.14·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.288 + 0.499i)3-s + 0.5·4-s + (−0.867 − 1.50i)5-s + (0.204 − 0.353i)6-s − 0.0574·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.613 + 1.06i)10-s + (0.133 + 0.231i)11-s + (−0.144 + 0.249i)12-s − 0.709·13-s + 0.0406·14-s + 1.00·15-s + 0.250·16-s + 1.49·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1338\)    =    \(2 \cdot 3 \cdot 223\)
Sign: $-0.997 + 0.0712i$
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: \((2,\ 1338,\ (\ :1/2),\ -0.997 + 0.0712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2565843288\)
\(L(\frac12)\) \(\approx\) \(0.2565843288\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.89 + 14.1i)T \)
good5 \( 1 + (1.93 + 3.35i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 0.151T + 7T^{2} \)
11 \( 1 + (-0.443 - 0.768i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.55T + 13T^{2} \)
17 \( 1 - 6.14T + 17T^{2} \)
19 \( 1 + (-1.72 + 2.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.21 + 3.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.88 + 6.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.208 - 0.360i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.08 - 1.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.19T + 41T^{2} \)
43 \( 1 + (2.79 - 4.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.30 + 9.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.48 - 4.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 + (7.54 - 13.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.36 - 7.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.92 + 5.06i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.302 - 0.524i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.25 - 3.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.87 + 4.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.67 - 8.09i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.11 - 1.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.382696766655458734959943416632, −8.407576758920523784619898709078, −7.86064398410774444142960109424, −7.00974914784021826253504371658, −5.72312921052478905336570476491, −4.93782811982024684020912310196, −4.20432884488372548186933815072, −3.02105119021876405426658328150, −1.30067724739835968969549871740, −0.15261201402289831726445057485, 1.58266930369195455381521603031, 3.04134754300483396894562674460, 3.51297896950889320226041984786, 5.19657551537875337951403271980, 6.19792852822337220311704126178, 6.97937909103164617424991882735, 7.65393642673750409161538301998, 7.968537560317635557052710314646, 9.333431039019519157246635674183, 10.07186346543878077273814678525

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