Properties

Label 2-7e2-7.4-c5-0-8
Degree $2$
Conductor $49$
Sign $-0.701 + 0.712i$
Analytic cond. $7.85880$
Root an. cond. $2.80335$
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  + (−4.13 − 7.16i)2-s + (−12.8 + 22.2i)3-s + (−18.2 + 31.5i)4-s + (14.3 + 24.8i)5-s + 212.·6-s + 37.0·8-s + (−207. − 359. i)9-s + (118. − 206. i)10-s + (135. − 233. i)11-s + (−467. − 810. i)12-s − 300.·13-s − 737.·15-s + (430. + 745. i)16-s + (306. − 530. i)17-s + (−1.71e3 + 2.97e3i)18-s + (−850. − 1.47e3i)19-s + ⋯
L(s)  = 1  + (−0.731 − 1.26i)2-s + (−0.822 + 1.42i)3-s + (−0.569 + 0.987i)4-s + (0.257 + 0.445i)5-s + 2.40·6-s + 0.204·8-s + (−0.853 − 1.47i)9-s + (0.376 − 0.651i)10-s + (0.336 − 0.582i)11-s + (−0.937 − 1.62i)12-s − 0.493·13-s − 0.846·15-s + (0.420 + 0.728i)16-s + (0.257 − 0.445i)17-s + (−1.24 + 2.16i)18-s + (−0.540 − 0.936i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.155761 - 0.371671i\)
\(L(\frac12)\) \(\approx\) \(0.155761 - 0.371671i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (4.13 + 7.16i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (12.8 - 22.2i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-14.3 - 24.8i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-135. + 233. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 300.T + 3.71e5T^{2} \)
17 \( 1 + (-306. + 530. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (850. + 1.47e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.59e3 + 2.76e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 4.29e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.01e3 + 1.75e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.57e3 + 4.46e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 7.14e3T + 1.15e8T^{2} \)
43 \( 1 + 1.95e4T + 1.47e8T^{2} \)
47 \( 1 + (-9.99e3 - 1.73e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.97e3 - 3.41e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.48e4 - 2.57e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.52e4 + 4.37e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.52e3 - 4.37e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.28e4T + 1.80e9T^{2} \)
73 \( 1 + (5.55e3 - 9.62e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (4.09e4 + 7.09e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.18e5T + 3.93e9T^{2} \)
89 \( 1 + (2.08e4 + 3.61e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 4.36e4T + 8.58e9T^{2} \)
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

−14.27737106192562477914760957585, −12.35013754304726603700649738953, −11.33814907849755039558740359197, −10.56678960346523774814755015914, −9.830818844885763225325918208625, −8.755090503870008332766651624401, −6.20915771758869122261751178939, −4.46734241398950283981021599740, −2.85392098088370950776717321527, −0.30634089578023981633279713377, 1.43330647653709002108681222743, 5.39347216826978821878781443128, 6.44672238455182007374662995395, 7.40119742411660567444640618487, 8.433299858104846219189301885838, 9.963715040187704196775598123146, 11.86764618237733337654374965477, 12.64386941917195467577004524815, 13.96567012507809796821178718248, 15.23299114552110668796979432471

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