Properties

Label 2-338-13.6-c2-0-19
Degree $2$
Conductor $338$
Sign $-0.261 + 0.965i$
Analytic cond. $9.20983$
Root an. cond. $3.03477$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 − 1.36i)2-s + (−1.16 + 2.01i)3-s + (−1.73 + i)4-s + (6.63 − 6.63i)5-s + (3.18 + 0.853i)6-s + (2.33 − 8.72i)7-s + (2 + 1.99i)8-s + (1.78 + 3.08i)9-s + (−11.4 − 6.63i)10-s + (−6.81 + 1.82i)11-s − 4.66i·12-s − 12.7·14-s + (5.66 + 21.1i)15-s + (1.99 − 3.46i)16-s + (9.62 − 5.55i)17-s + (3.56 − 3.56i)18-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.388 + 0.673i)3-s + (−0.433 + 0.250i)4-s + (1.32 − 1.32i)5-s + (0.530 + 0.142i)6-s + (0.333 − 1.24i)7-s + (0.250 + 0.249i)8-s + (0.197 + 0.342i)9-s + (−1.14 − 0.663i)10-s + (−0.619 + 0.165i)11-s − 0.388i·12-s − 0.912·14-s + (0.377 + 1.40i)15-s + (0.124 − 0.216i)16-s + (0.565 − 0.326i)17-s + (0.197 − 0.197i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.261 + 0.965i$
Analytic conductor: \(9.20983\)
Root analytic conductor: \(3.03477\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1),\ -0.261 + 0.965i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.923789 - 1.20698i\)
\(L(\frac12)\) \(\approx\) \(0.923789 - 1.20698i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
13 \( 1 \)
good3 \( 1 + (1.16 - 2.01i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-6.63 + 6.63i)T - 25iT^{2} \)
7 \( 1 + (-2.33 + 8.72i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (6.81 - 1.82i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (-9.62 + 5.55i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (12.9 + 3.46i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (-6.35 - 3.66i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-0.105 + 0.183i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-28.2 + 28.2i)T - 961iT^{2} \)
37 \( 1 + (6.11 - 1.63i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (20.2 + 75.5i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-21.1 + 12.1i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-3.04 - 3.04i)T + 2.20e3iT^{2} \)
53 \( 1 - 22.3T + 2.80e3T^{2} \)
59 \( 1 + (15.2 - 56.7i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-10.8 - 18.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (0.809 + 3.02i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (106. + 28.5i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-88.1 - 88.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 45.8T + 6.24e3T^{2} \)
83 \( 1 + (61.6 - 61.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (-119. + 32.0i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-78.2 - 20.9i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−10.63749669789304020150775830253, −10.27365479872624871327911038185, −9.522897359156228519651906070417, −8.533140466472080507006793522855, −7.41540191455120467862742951444, −5.70510282422156041960634349390, −4.87309981297862442483583623941, −4.16144605067120087865000230907, −2.13487347933567708814898707499, −0.824968492323831537333238936786, 1.74046768916046332977154524309, 2.96577506835512798708020913673, 5.18471973425810479585640041583, 6.12408454205034061519457732573, 6.46572153747584353712659424433, 7.60204319232580743954938733626, 8.729665878241587483364468138861, 9.771065236788645008155916602660, 10.50469420401933593536277099039, 11.58017959539247624667417550571

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