Properties

Label 2-234-13.7-c2-0-0
Degree $2$
Conductor $234$
Sign $-0.0616 - 0.998i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (−4.02 − 4.02i)5-s + (−4.99 − 1.33i)7-s + (−1.99 + 2i)8-s + (6.97 + 4.02i)10-s + (3.41 + 12.7i)11-s + (3.81 + 12.4i)13-s + 7.31·14-s + (1.99 − 3.46i)16-s + (16.3 − 9.42i)17-s + (−7.85 + 29.3i)19-s + (−10.9 − 2.94i)20-s + (−9.31 − 16.1i)22-s + (10.4 + 6.05i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.250i)4-s + (−0.805 − 0.805i)5-s + (−0.714 − 0.191i)7-s + (−0.249 + 0.250i)8-s + (0.697 + 0.402i)10-s + (0.310 + 1.15i)11-s + (0.293 + 0.956i)13-s + 0.522·14-s + (0.124 − 0.216i)16-s + (0.960 − 0.554i)17-s + (−0.413 + 1.54i)19-s + (−0.549 − 0.147i)20-s + (−0.423 − 0.733i)22-s + (0.455 + 0.263i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0616 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0616 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.0616 - 0.998i$
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: \((2,\ 234,\ (\ :1),\ -0.0616 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.427412 + 0.454624i\)
\(L(\frac12)\) \(\approx\) \(0.427412 + 0.454624i\)
\(L(2)\) 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 \( 1 + (4.02 + 4.02i)T + 25iT^{2} \)
7 \( 1 + (4.99 + 1.33i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-3.41 - 12.7i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (-16.3 + 9.42i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (7.85 - 29.3i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-10.4 - 6.05i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (18.4 - 31.9i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (6.48 + 6.48i)T + 961iT^{2} \)
37 \( 1 + (-13.6 - 50.7i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-6.38 + 1.70i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-11.7 + 6.80i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (61.6 - 61.6i)T - 2.20e3iT^{2} \)
53 \( 1 + 4.64T + 2.80e3T^{2} \)
59 \( 1 + (24.1 + 6.45i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (19.2 + 33.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-91.8 + 24.6i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (10.5 - 39.2i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-60.4 + 60.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 94.2T + 6.24e3T^{2} \)
83 \( 1 + (-63.4 - 63.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (40.3 + 150. i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (39.2 - 146. i)T + (-8.14e3 - 4.70e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.22009005173516769714844395869, −11.28050762980918764457521778118, −9.936979268543481095750419775270, −9.370334655381302869722496836713, −8.245191987535843147723557336813, −7.37216928662659368613497791399, −6.35002684974124465980984525779, −4.82498663089432483505525825056, −3.59697364331128226123984377687, −1.45935262065990165795701437996, 0.44345621525577747729013363114, 2.87342634946544395069420094270, 3.68263729606154730148602233378, 5.77300126731020162006500388282, 6.79837483880655847082199778088, 7.82430608839237826974355730715, 8.722963287692863371785507815389, 9.799090764379678593899236852545, 10.94428054722730972178771646356, 11.27890215353182321622152975636

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