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Properties

Label 2-338-13.9-c1-0-7
Degree $2$
Conductor $338$
Sign $0.997 + 0.0743i$
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 + (1.02 − 1.77i)3-s + (−0.499 − 0.866i)4-s + 3.60·5-s + (1.02 + 1.77i)6-s + (0.554 + 0.961i)7-s + 0.999·8-s + (−0.599 − 1.03i)9-s + (−1.80 + 3.12i)10-s + (−1.17 + 2.04i)11-s − 2.04·12-s − 1.10·14-s + (3.69 − 6.39i)15-s + (−0.5 + 0.866i)16-s + (−2.98 − 5.16i)17-s + 1.19·18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.591 − 1.02i)3-s + (−0.249 − 0.433i)4-s + 1.61·5-s + (0.418 + 0.724i)6-s + (0.209 + 0.363i)7-s + 0.353·8-s + (−0.199 − 0.345i)9-s + (−0.569 + 0.986i)10-s + (−0.355 + 0.615i)11-s − 0.591·12-s − 0.296·14-s + (0.953 − 1.65i)15-s + (−0.125 + 0.216i)16-s + (−0.722 − 1.25i)17-s + 0.282·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.997 + 0.0743i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ 0.997 + 0.0743i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65479 - 0.0615708i\)
\(L(\frac12)\) \(\approx\) \(1.65479 - 0.0615708i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.02 + 1.77i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3.60T + 5T^{2} \)
7 \( 1 + (-0.554 - 0.961i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.17 - 2.04i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.98 + 5.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.455 - 0.789i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.69 - 2.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.89 + 3.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.49T + 31T^{2} \)
37 \( 1 + (-2.44 + 4.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.59 - 6.22i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.257 - 0.446i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.98T + 47T^{2} \)
53 \( 1 + 3.38T + 53T^{2} \)
59 \( 1 + (-5.07 - 8.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.219 - 0.380i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.07 + 1.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.307 - 0.533i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.32T + 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 - 0.911T + 83T^{2} \)
89 \( 1 + (-1.87 + 3.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.33 + 12.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.58305799335539460224334873305, −10.23632857883254612241261229323, −9.485379916625784819233904704045, −8.724481374705352009394083479267, −7.60882291488602046818231503413, −6.87620157490424999062029657279, −5.86799428342797826067034954372, −4.93363652332603313301164947944, −2.52376663183322469146357626148, −1.68535798131608311633509970104, 1.80460199138052354226772363009, 3.07889329322552098974641352159, 4.27830883471772750178638811365, 5.47480828643883784025301631126, 6.66741583509625049523913822197, 8.337414027566473240431500669193, 8.990231605949742310400338591334, 9.811578882082388345163294182280, 10.47193326686457439067800459558, 11.01640953541399808360644567366

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