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Properties

Label 2-268-67.64-c1-0-4
Degree $2$
Conductor $268$
Sign $-0.685 + 0.728i$
Analytic cond. $2.13999$
Root an. cond. $1.46287$
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  + (0.300 − 2.09i)3-s + (−0.604 − 1.32i)5-s + (−3.15 − 0.925i)7-s + (−1.40 − 0.412i)9-s + (−0.624 − 1.36i)11-s + (−0.450 + 0.520i)13-s + (−2.94 + 0.866i)15-s + (1.67 − 1.07i)17-s + (−0.854 + 0.250i)19-s + (−2.88 + 6.30i)21-s + (−0.184 + 1.28i)23-s + (1.88 − 2.17i)25-s + (1.34 − 2.95i)27-s + 4.85·29-s + (−4.43 − 5.11i)31-s + ⋯
L(s)  = 1  + (0.173 − 1.20i)3-s + (−0.270 − 0.591i)5-s + (−1.19 − 0.349i)7-s + (−0.468 − 0.137i)9-s + (−0.188 − 0.412i)11-s + (−0.124 + 0.144i)13-s + (−0.761 + 0.223i)15-s + (0.405 − 0.260i)17-s + (−0.196 + 0.0575i)19-s + (−0.628 + 1.37i)21-s + (−0.0385 + 0.267i)23-s + (0.377 − 0.435i)25-s + (0.259 − 0.568i)27-s + 0.902·29-s + (−0.796 − 0.919i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(268\)    =    \(2^{2} \cdot 67\)
Sign: $-0.685 + 0.728i$
Analytic conductor: \(2.13999\)
Root analytic conductor: \(1.46287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{268} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 268,\ (\ :1/2),\ -0.685 + 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.404477 - 0.936030i\)
\(L(\frac12)\) \(\approx\) \(0.404477 - 0.936030i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
67 \( 1 + (2.59 + 7.76i)T \)
good3 \( 1 + (-0.300 + 2.09i)T + (-2.87 - 0.845i)T^{2} \)
5 \( 1 + (0.604 + 1.32i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (3.15 + 0.925i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (0.624 + 1.36i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.450 - 0.520i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-1.67 + 1.07i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (0.854 - 0.250i)T + (15.9 - 10.2i)T^{2} \)
23 \( 1 + (0.184 - 1.28i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 - 4.85T + 29T^{2} \)
31 \( 1 + (4.43 + 5.11i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 - 3.34T + 37T^{2} \)
41 \( 1 + (1.41 - 0.908i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-9.18 + 5.90i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + (-0.284 + 1.97i)T + (-45.0 - 13.2i)T^{2} \)
53 \( 1 + (-7.92 - 5.09i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-6.65 - 7.68i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (4.37 - 9.58i)T + (-39.9 - 46.1i)T^{2} \)
71 \( 1 + (-0.631 - 0.406i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-0.459 + 1.00i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (0.351 - 0.405i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-6.78 - 14.8i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-1.76 - 12.2i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 - 1.69T + 97T^{2} \)
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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

−12.03084845650224159178364832828, −10.64777997240273423576477497971, −9.577815867129677670234403462593, −8.542906407964978461496940239295, −7.56576840615267124337808304834, −6.76199474060078417203466460149, −5.73586882658719464637473121219, −4.09773967885914383287620792442, −2.63209044237223002023253737503, −0.78733716067798142682662128841, 2.86552439396494443501908542796, 3.74290250420127657391584828853, 4.98360194202755348677791449100, 6.28663038698475155260339688700, 7.32695534107883605468899895855, 8.731995255302587022723323074070, 9.621366930321956161289467539706, 10.26596250888681615210695613045, 11.04265348528601553959129661471, 12.34632184148404919497183406517

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