↑

Properties

Label 2-84-7.2-c5-0-6
Degree $2$
Conductor $84$
Sign $-0.349 + 0.937i$
Analytic cond. $13.4722$
Root an. cond. $3.67045$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.5 + 7.79i)3-s + (46.4 − 80.3i)5-s + (−118. − 52.8i)7-s + (−40.5 + 70.1i)9-s + (−70.3 − 121. i)11-s − 1.11e3·13-s + 835.·15-s + (−27.4 − 47.5i)17-s + (855. − 1.48e3i)19-s + (−120. − 1.16e3i)21-s + (1.64e3 − 2.84e3i)23-s + (−2.74e3 − 4.75e3i)25-s − 729·27-s − 3.79e3·29-s + (2.42e3 + 4.19e3i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.830 − 1.43i)5-s + (−0.913 − 0.407i)7-s + (−0.166 + 0.288i)9-s + (−0.175 − 0.303i)11-s − 1.82·13-s + 0.958·15-s + (−0.0230 − 0.0398i)17-s + (0.543 − 0.942i)19-s + (−0.0596 − 0.574i)21-s + (0.647 − 1.12i)23-s + (−0.878 − 1.52i)25-s − 0.192·27-s − 0.837·29-s + (0.452 + 0.784i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 + 0.937i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.349 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.349 + 0.937i$
Analytic conductor: \(13.4722\)
Root analytic conductor: \(3.67045\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :5/2),\ -0.349 + 0.937i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.785765 - 1.13138i\)
\(L(\frac12)\) \(\approx\) \(0.785765 - 1.13138i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-4.5 - 7.79i)T \)
7 \( 1 + (118. + 52.8i)T \)
good5 \( 1 + (-46.4 + 80.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (70.3 + 121. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 1.11e3T + 3.71e5T^{2} \)
17 \( 1 + (27.4 + 47.5i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-855. + 1.48e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-1.64e3 + 2.84e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 3.79e3T + 2.05e7T^{2} \)
31 \( 1 + (-2.42e3 - 4.19e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.68e3 + 9.84e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.03e4T + 1.15e8T^{2} \)
43 \( 1 - 7.13e3T + 1.47e8T^{2} \)
47 \( 1 + (-8.20e3 + 1.42e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.04e4 - 1.81e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.81e4 - 3.13e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.47e3 - 4.28e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.14e4 - 1.98e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 2.63e4T + 1.80e9T^{2} \)
73 \( 1 + (2.76e4 + 4.79e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-2.49e4 + 4.32e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 4.48e4T + 3.93e9T^{2} \)
89 \( 1 + (6.39e4 - 1.10e5i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 6.56e4T + 8.58e9T^{2} \)
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

−13.02741430687984908008012757150, −12.19068378761526755133923235874, −10.41363733177450411376342760542, −9.504488548281649637739702314764, −8.842014451151671326518149548054, −7.19406879819791740366265629200, −5.49814810114195602304991809387, −4.51771336869341740947453197125, −2.60358281801876229093346292117, −0.51937325190237593135441804386, 2.20108013975625415349399278912, 3.16434361400646930423604786594, 5.60434048375190981678367947790, 6.75979001434597726818984506862, 7.59296479946980574305069520869, 9.610602569151661042152575981513, 9.971235118039912962144125961376, 11.54273966427230089146447707697, 12.68579488606645178240679712813, 13.67441992725665850545295592024

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