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Properties

Label 2-338-13.10-c1-0-8
Degree $2$
Conductor $338$
Sign $0.839 - 0.543i$
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 + (1.5 + 2.59i)3-s + (0.499 − 0.866i)4-s − i·5-s + (2.59 + 1.5i)6-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (−3 + 5.19i)9-s + (−0.5 − 0.866i)10-s + (1.73 − i)11-s + 3·12-s + 0.999·14-s + (2.59 − 1.5i)15-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + 6i·18-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.866 + 1.49i)3-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (1.06 + 0.612i)6-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (−1 + 1.73i)9-s + (−0.158 − 0.273i)10-s + (0.522 − 0.301i)11-s + 0.866·12-s + 0.267·14-s + (0.670 − 0.387i)15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + 1.41i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ 0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28401 + 0.675586i\)
\(L(\frac12)\) \(\approx\) \(2.28401 + 0.675586i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + iT - 5T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.73 + i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.19 + 3i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (2.59 - 1.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 13iT - 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (8.66 + 5i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.33 - 2.5i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (5.19 - 3i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.1 + 7i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.43911695887120971118537771681, −10.64515890454952294155269271635, −9.913185870502105458090272301155, −8.709542587015666102174173117482, −8.512768303978593408171527276184, −6.60071325138595913732005757206, −5.19701047741107606152159079311, −4.42488536811783229246609145604, −3.57961127437682844819302955625, −2.26087770069826664507901085731, 1.74981381815733341569202639170, 2.91353835114745998091092565214, 4.21412522403204438203093364766, 5.91541864476566554398979113211, 6.82805544314088257573520170084, 7.44753880197435840843448491105, 8.316422503449317614447196602139, 9.256399762484528168308855329567, 10.78814160458541141660259805470, 11.86141300282434313190287599116

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