↑

Properties

Label 2-56e2-448.381-c0-0-0
Degree $2$
Conductor $3136$
Sign $-0.986 - 0.160i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.608 + 0.793i)2-s + (−0.258 − 0.965i)4-s + (0.923 + 0.382i)8-s + (0.130 + 0.991i)9-s + (−0.357 + 1.05i)11-s + (−0.866 + 0.499i)16-s + (−0.866 − 0.499i)18-s + (−0.617 − 0.923i)22-s + (−1.83 + 0.241i)23-s + (−0.991 − 0.130i)25-s + (−1.63 − 0.324i)29-s + (0.130 − 0.991i)32-s + (0.923 − 0.382i)36-s + (1.95 + 0.128i)37-s + (0.382 + 1.92i)43-s + (1.10 + 0.0726i)44-s + ⋯
L(s)  = 1  + (−0.608 + 0.793i)2-s + (−0.258 − 0.965i)4-s + (0.923 + 0.382i)8-s + (0.130 + 0.991i)9-s + (−0.357 + 1.05i)11-s + (−0.866 + 0.499i)16-s + (−0.866 − 0.499i)18-s + (−0.617 − 0.923i)22-s + (−1.83 + 0.241i)23-s + (−0.991 − 0.130i)25-s + (−1.63 − 0.324i)29-s + (0.130 − 0.991i)32-s + (0.923 − 0.382i)36-s + (1.95 + 0.128i)37-s + (0.382 + 1.92i)43-s + (1.10 + 0.0726i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $-0.986 - 0.160i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (3069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :0),\ -0.986 - 0.160i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5102005650\)
\(L(\frac12)\) \(\approx\) \(0.5102005650\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.608 - 0.793i)T \)
7 \( 1 \)
good3 \( 1 + (-0.130 - 0.991i)T^{2} \)
5 \( 1 + (0.991 + 0.130i)T^{2} \)
11 \( 1 + (0.357 - 1.05i)T + (-0.793 - 0.608i)T^{2} \)
13 \( 1 + (-0.382 + 0.923i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.608 + 0.793i)T^{2} \)
23 \( 1 + (1.83 - 0.241i)T + (0.965 - 0.258i)T^{2} \)
29 \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.95 - 0.128i)T + (0.991 + 0.130i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.382 - 1.92i)T + (-0.923 + 0.382i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.125 + 0.369i)T + (-0.793 - 0.608i)T^{2} \)
59 \( 1 + (-0.608 + 0.793i)T^{2} \)
61 \( 1 + (-0.793 + 0.608i)T^{2} \)
67 \( 1 + (0.293 + 0.257i)T + (0.130 + 0.991i)T^{2} \)
71 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.258 + 0.965i)T^{2} \)
79 \( 1 + (0.739 + 0.198i)T + (0.866 + 0.5i)T^{2} \)
83 \( 1 + (0.382 - 0.923i)T^{2} \)
89 \( 1 + (0.258 + 0.965i)T^{2} \)
97 \( 1 + T^{2} \)
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

−9.321948906979247177324313143633, −8.115198658248500682311022929463, −7.79077355480019854807002030026, −7.27708278392494016727819891288, −6.13437934769713606684753709318, −5.67725611452597308098614596682, −4.65503919473487236533955302684, −4.11713474933476416046843123716, −2.38817479921352734630180896736, −1.68073617721570836706071640383, 0.36774626138106309137798120445, 1.73291337178893938356373000102, 2.75604033321526942030090720843, 3.75066453878983379971136304074, 4.14366274525825851213138511284, 5.60999236272982366173064359213, 6.17908893156323369770801927088, 7.31960212241121352735628585996, 7.888795524032306358430315933422, 8.677931565806529070525932399041

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