↑

Properties

Label 2-28e2-112.53-c1-0-66
Degree $2$
Conductor $784$
Sign $-0.317 + 0.948i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.164 + 1.40i)2-s + (−0.338 − 1.26i)3-s + (−1.94 + 0.461i)4-s + (0.958 − 3.57i)5-s + (1.72 − 0.683i)6-s + (−0.968 − 2.65i)8-s + (1.11 − 0.642i)9-s + (5.17 + 0.757i)10-s + (−4.80 + 1.28i)11-s + (1.24 + 2.30i)12-s + (3.14 − 3.14i)13-s − 4.84·15-s + (3.57 − 1.79i)16-s + (−3.33 + 5.76i)17-s + (1.08 + 1.45i)18-s + (−4.73 − 1.26i)19-s + ⋯
L(s)  = 1  + (0.116 + 0.993i)2-s + (−0.195 − 0.730i)3-s + (−0.972 + 0.230i)4-s + (0.428 − 1.59i)5-s + (0.702 − 0.279i)6-s + (−0.342 − 0.939i)8-s + (0.371 − 0.214i)9-s + (1.63 + 0.239i)10-s + (−1.44 + 0.388i)11-s + (0.358 + 0.665i)12-s + (0.871 − 0.871i)13-s − 1.25·15-s + (0.893 − 0.449i)16-s + (−0.807 + 1.39i)17-s + (0.256 + 0.343i)18-s + (−1.08 − 0.291i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.317 + 0.948i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.317 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.533854 - 0.742029i\)
\(L(\frac12)\) \(\approx\) \(0.533854 - 0.742029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.164 - 1.40i)T \)
7 \( 1 \)
good3 \( 1 + (0.338 + 1.26i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.958 + 3.57i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (4.80 - 1.28i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.14 + 3.14i)T - 13iT^{2} \)
17 \( 1 + (3.33 - 5.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.73 + 1.26i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.83 - 1.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0287 - 0.0287i)T - 29iT^{2} \)
31 \( 1 + (-2.29 + 3.97i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.343 + 1.28i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.49iT - 41T^{2} \)
43 \( 1 + (0.947 + 0.947i)T + 43iT^{2} \)
47 \( 1 + (-0.399 - 0.691i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.83 + 0.490i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.24 + 1.40i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.52 + 1.47i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.83 + 10.5i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.16iT - 71T^{2} \)
73 \( 1 + (1.13 + 0.654i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.166 + 0.287i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.9 + 10.9i)T - 83iT^{2} \)
89 \( 1 + (-5.63 + 3.25i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.56T + 97T^{2} \)
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

−9.854029224286122264512816679675, −8.796293723451518175468537191975, −8.234852162655366243827510719857, −7.60826888620540714961468060317, −6.30093924231266382224594167023, −5.80381505500826608795909717053, −4.83056293191055037878600439875, −3.99060133497389614175294516986, −1.88259420780017046313487788344, −0.43972369373390238968722834329, 2.13228120557296723554934971406, 2.92202345525662109506065483611, 4.01738738354485593430483426715, 4.94012755047491270144830468935, 6.02174174685319672131463425509, 6.97340080051241831089349183386, 8.185910543930400294442108127545, 9.237361895587245653419293116302, 10.11629837884717531706913127039, 10.60767916094255058815504735849

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