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Properties

Label 2-234-13.7-c2-0-2
Degree $2$
Conductor $234$
Sign $0.508 - 0.860i$
Analytic cond. $6.37603$
Root an. cond. $2.52508$
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  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (6.39 + 6.39i)5-s + (9.23 + 2.47i)7-s + (−1.99 + 2i)8-s + (−11.0 − 6.39i)10-s + (−4.21 − 15.7i)11-s + (−3.81 + 12.4i)13-s − 13.5·14-s + (1.99 − 3.46i)16-s + (−8.31 + 4.80i)17-s + (2.56 − 9.56i)19-s + (17.4 + 4.67i)20-s + (11.5 + 19.9i)22-s + (23.6 + 13.6i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.250i)4-s + (1.27 + 1.27i)5-s + (1.31 + 0.353i)7-s + (−0.249 + 0.250i)8-s + (−1.10 − 0.639i)10-s + (−0.383 − 1.43i)11-s + (−0.293 + 0.956i)13-s − 0.965·14-s + (0.124 − 0.216i)16-s + (−0.489 + 0.282i)17-s + (0.134 − 0.503i)19-s + (0.873 + 0.233i)20-s + (0.523 + 0.906i)22-s + (1.03 + 0.594i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.508 - 0.860i$
Analytic conductor: \(6.37603\)
Root analytic conductor: \(2.52508\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1),\ 0.508 - 0.860i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.34407 + 0.766972i\)
\(L(\frac12)\) \(\approx\) \(1.34407 + 0.766972i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 \)
13 \( 1 + (3.81 - 12.4i)T \)
good5 \( 1 + (-6.39 - 6.39i)T + 25iT^{2} \)
7 \( 1 + (-9.23 - 2.47i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (4.21 + 15.7i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (8.31 - 4.80i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-2.56 + 9.56i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-23.6 - 13.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-13.8 + 23.9i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-11.5 - 11.5i)T + 961iT^{2} \)
37 \( 1 + (5.45 + 20.3i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (39.1 - 10.4i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (30.9 - 17.8i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (25.5 - 25.5i)T - 2.20e3iT^{2} \)
53 \( 1 - 39.0T + 2.80e3T^{2} \)
59 \( 1 + (-46.0 - 12.3i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (35.2 + 61.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-38.7 + 10.3i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-8.28 + 30.9i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (9.68 - 9.68i)T - 5.32e3iT^{2} \)
79 \( 1 + 56.1T + 6.24e3T^{2} \)
83 \( 1 + (4.59 + 4.59i)T + 6.88e3iT^{2} \)
89 \( 1 + (29.9 + 111. i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (9.04 - 33.7i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−11.46121559764219297134285743321, −11.16428604690260837796905756086, −10.21420727602716579258135986706, −9.167853375828692012417517077953, −8.286296676618028572883758921721, −7.03538667756282915479536917619, −6.15873395807022388032548617451, −5.12461938524940214017057509128, −2.88980307127751353882995170725, −1.75076330201248741363593462847, 1.19076170691222236621621387002, 2.24518868273402475931599676746, 4.72652679573873449241558269414, 5.28136949216529462716764687323, 6.94844897043527529311193724170, 8.113771154749720607682315024525, 8.835225947161455076967997554122, 9.962832870648237053728241002624, 10.44523296253669658191242628265, 11.83141325327791427057265317373

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