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Properties

Label 2-338-13.4-c1-0-8
Degree $2$
Conductor $338$
Sign $0.824 - 0.565i$
Analytic cond. $2.69894$
Root an. cond. $1.64284$
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  + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s − 3i·5-s + (−0.866 + 0.499i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (1 + 1.73i)9-s + (1.5 − 2.59i)10-s + (5.19 + 3i)11-s − 0.999·12-s + 0.999·14-s + (2.59 + 1.5i)15-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + 2i·18-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s − 1.34i·5-s + (−0.353 + 0.204i)6-s + (0.327 − 0.188i)7-s + 0.353i·8-s + (0.333 + 0.577i)9-s + (0.474 − 0.821i)10-s + (1.56 + 0.904i)11-s − 0.288·12-s + 0.267·14-s + (0.670 + 0.387i)15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + 0.471i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.824 - 0.565i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ 0.824 - 0.565i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78073 + 0.551653i\)
\(L(\frac12)\) \(\approx\) \(1.78073 + 0.551653i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3iT - 5T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.19 - 3i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.73 + i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (6.06 + 3.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-5.19 + 3i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.1 + 7i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.59 - 1.5i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + (5.19 + 3i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.66 - 5i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.82886036140134636675036524714, −10.90404238558420770391334321437, −9.559964482168443674386348551356, −8.975343567596513651595160161131, −7.72437897160471159367465389417, −6.77929654050765462482870108383, −5.33892699319055214569349783933, −4.69734735241645481312565641433, −3.93672797231837237755924166784, −1.66997696352796382074964464652, 1.55653915841502414727331083642, 3.18528608724977792782108047105, 4.06627112795307223189785650988, 5.85189780807458816715704525489, 6.49148667943659092833895167288, 7.22580839930106836723591290574, 8.697849266539438259622865848175, 9.850147166656173589126068592679, 10.84531898189319695833100174547, 11.62011912322396610554581162090

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