↑

Properties

Label 2-7e2-49.29-c1-0-1
Degree $2$
Conductor $49$
Sign $0.926 - 0.376i$
Analytic cond. $0.391266$
Root an. cond. $0.625513$
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  + (0.658 + 0.317i)2-s + (0.255 + 1.12i)3-s + (−0.914 − 1.14i)4-s + (−0.0575 − 0.252i)5-s + (−0.186 + 0.818i)6-s + (−2.16 + 1.51i)7-s + (−0.563 − 2.46i)8-s + (1.51 − 0.728i)9-s + (0.0420 − 0.184i)10-s + (−3.38 − 1.63i)11-s + (1.05 − 1.31i)12-s + (2.38 + 1.15i)13-s + (−1.90 + 0.309i)14-s + (0.267 − 0.129i)15-s + (−0.240 + 1.05i)16-s + (−3.20 + 4.01i)17-s + ⋯
L(s)  = 1  + (0.465 + 0.224i)2-s + (0.147 + 0.647i)3-s + (−0.457 − 0.573i)4-s + (−0.0257 − 0.112i)5-s + (−0.0762 + 0.334i)6-s + (−0.819 + 0.572i)7-s + (−0.199 − 0.872i)8-s + (0.504 − 0.242i)9-s + (0.0132 − 0.0582i)10-s + (−1.02 − 0.491i)11-s + (0.303 − 0.380i)12-s + (0.662 + 0.319i)13-s + (−0.509 + 0.0828i)14-s + (0.0691 − 0.0333i)15-s + (−0.0601 + 0.263i)16-s + (−0.777 + 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.926 - 0.376i$
Analytic conductor: \(0.391266\)
Root analytic conductor: \(0.625513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :1/2),\ 0.926 - 0.376i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904243 + 0.176795i\)
\(L(\frac12)\) \(\approx\) \(0.904243 + 0.176795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.16 - 1.51i)T \)
good2 \( 1 + (-0.658 - 0.317i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (-0.255 - 1.12i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (0.0575 + 0.252i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (3.38 + 1.63i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-2.38 - 1.15i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (3.20 - 4.01i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 - 3.83T + 19T^{2} \)
23 \( 1 + (-0.752 - 0.944i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-0.581 + 0.728i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + 4.29T + 31T^{2} \)
37 \( 1 + (-2.52 + 3.16i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (1.90 + 8.34i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (1.08 - 4.75i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (10.7 + 5.19i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (2.39 + 2.99i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-1.19 + 5.24i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (8.51 - 10.6i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 + (-4.94 - 6.19i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-7.59 + 3.65i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + 8.64T + 79T^{2} \)
83 \( 1 + (-10.3 + 4.98i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (6.56 - 3.16i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + 8.15T + 97T^{2} \)
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

−15.70550956577634211027263828099, −14.76865318995075484965644099150, −13.40307911984280780160871273315, −12.69073624814667930799576980001, −10.78870900361812946900418867743, −9.719921649731390927198032095467, −8.723171746318645885120613487316, −6.52384180223352683596250694816, −5.22684745679568487389320782640, −3.67569552583005435813613514120, 3.04596108039067770554766221350, 4.83606670085661235434745952908, 6.93431649627111701680729515848, 7.989135870775891489433939975740, 9.601815353858809812564004837502, 11.06717506551054141758561081788, 12.63569710355074804059098232789, 13.16249422692296953285757708231, 13.92424817011327426186061041252, 15.61390485064024734271331383877

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