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Properties

Label 2-338-13.3-c1-0-0
Degree $2$
Conductor $338$
Sign $0.668 - 0.743i$
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  + (−0.5 − 0.866i)2-s + (−1.34 − 2.33i)3-s + (−0.499 + 0.866i)4-s − 2.49·5-s + (−1.34 + 2.33i)6-s + (−0.801 + 1.38i)7-s + 0.999·8-s + (−2.12 + 3.67i)9-s + (1.24 + 2.15i)10-s + (1.02 + 1.77i)11-s + 2.69·12-s + 1.60·14-s + (3.35 + 5.81i)15-s + (−0.5 − 0.866i)16-s + (2.27 − 3.93i)17-s + 4.24·18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.777 − 1.34i)3-s + (−0.249 + 0.433i)4-s − 1.11·5-s + (−0.549 + 0.951i)6-s + (−0.303 + 0.524i)7-s + 0.353·8-s + (−0.707 + 1.22i)9-s + (0.394 + 0.682i)10-s + (0.308 + 0.535i)11-s + 0.777·12-s + 0.428·14-s + (0.866 + 1.50i)15-s + (−0.125 − 0.216i)16-s + (0.550 − 0.954i)17-s + 1.00·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.668 - 0.743i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ 0.668 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.199254 + 0.0887787i\)
\(L(\frac12)\) \(\approx\) \(0.199254 + 0.0887787i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good3 \( 1 + (1.34 + 2.33i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.49T + 5T^{2} \)
7 \( 1 + (0.801 - 1.38i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.02 - 1.77i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.27 + 3.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.42 - 4.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.35 + 2.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.60 - 7.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.10T + 31T^{2} \)
37 \( 1 + (-3.80 - 6.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.73 + 3.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.67 - 9.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.219T + 47T^{2} \)
53 \( 1 + 2.71T + 53T^{2} \)
59 \( 1 + (2.03 - 3.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.20 - 9.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.03 + 10.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.643 + 1.11i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.62T + 73T^{2} \)
79 \( 1 + 5.32T + 79T^{2} \)
83 \( 1 + 4.85T + 83T^{2} \)
89 \( 1 + (8.28 + 14.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.32 + 4.01i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.89199898803159579928755322109, −11.10617692330751339115597735167, −9.974997708162701247484703724197, −8.707851406706358558484866683884, −7.77836277377810063946822940706, −7.07462743564430799668570133378, −6.01600840931380792474115826700, −4.60142643629025004703552239143, −3.07159819627641704994106826823, −1.47598106817497577183703762647, 0.19928479515654263867817178747, 3.70003615021740839695636772543, 4.27761746080295665874480269363, 5.49884868430415157924867057342, 6.49003965557592497922227303896, 7.67939258043849700698785599589, 8.642322206654627615378689762840, 9.671036998651045945456310909901, 10.48456845005493746519734071656, 11.19502938430769246151381101932

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