Properties

Label 2-7e2-49.2-c5-0-20
Degree $2$
Conductor $49$
Sign $-0.532 + 0.846i$
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  + (5.22 + 1.61i)2-s + (8.72 − 22.2i)3-s + (−1.77 − 1.20i)4-s + (−104. + 15.7i)5-s + (81.3 − 102. i)6-s + (122. − 42.4i)7-s + (−116. − 145. i)8-s + (−240. − 223. i)9-s + (−572. − 86.2i)10-s + (99.8 − 92.6i)11-s + (−42.3 + 28.8i)12-s + (62.1 + 272. i)13-s + (707. − 24.4i)14-s + (−563. + 2.46e3i)15-s + (−347. − 885. i)16-s + (−58.6 − 783. i)17-s + ⋯
L(s)  = 1  + (0.923 + 0.284i)2-s + (0.559 − 1.42i)3-s + (−0.0553 − 0.0377i)4-s + (−1.87 + 0.282i)5-s + (0.923 − 1.15i)6-s + (0.944 − 0.327i)7-s + (−0.642 − 0.805i)8-s + (−0.989 − 0.917i)9-s + (−1.80 − 0.272i)10-s + (0.248 − 0.230i)11-s + (−0.0848 + 0.0578i)12-s + (0.101 + 0.446i)13-s + (0.965 − 0.0333i)14-s + (−0.646 + 2.83i)15-s + (−0.339 − 0.864i)16-s + (−0.0492 − 0.657i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 + 0.846i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.532 + 0.846i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -0.532 + 0.846i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.943854 - 1.70838i\)
\(L(\frac12)\) \(\approx\) \(0.943854 - 1.70838i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-122. + 42.4i)T \)
good2 \( 1 + (-5.22 - 1.61i)T + (26.4 + 18.0i)T^{2} \)
3 \( 1 + (-8.72 + 22.2i)T + (-178. - 165. i)T^{2} \)
5 \( 1 + (104. - 15.7i)T + (2.98e3 - 921. i)T^{2} \)
11 \( 1 + (-99.8 + 92.6i)T + (1.20e4 - 1.60e5i)T^{2} \)
13 \( 1 + (-62.1 - 272. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (58.6 + 783. i)T + (-1.40e6 + 2.11e5i)T^{2} \)
19 \( 1 + (-1.38e3 + 2.39e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (236. - 3.15e3i)T + (-6.36e6 - 9.59e5i)T^{2} \)
29 \( 1 + (-4.71e3 - 2.26e3i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (2.55e3 + 4.42e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (1.88e3 - 1.28e3i)T + (2.53e7 - 6.45e7i)T^{2} \)
41 \( 1 + (5.79e3 + 7.26e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (8.40e3 - 1.05e4i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (4.89e3 + 1.51e3i)T + (1.89e8 + 1.29e8i)T^{2} \)
53 \( 1 + (-1.21e4 - 8.26e3i)T + (1.52e8 + 3.89e8i)T^{2} \)
59 \( 1 + (7.34e3 + 1.10e3i)T + (6.83e8 + 2.10e8i)T^{2} \)
61 \( 1 + (-9.82e3 + 6.69e3i)T + (3.08e8 - 7.86e8i)T^{2} \)
67 \( 1 + (-4.96e3 - 8.59e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (2.05e4 - 9.90e3i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (1.38e4 - 4.27e3i)T + (1.71e9 - 1.16e9i)T^{2} \)
79 \( 1 + (-4.31e4 + 7.46e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-492. + 2.15e3i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (2.30e3 + 2.13e3i)T + (4.17e8 + 5.56e9i)T^{2} \)
97 \( 1 - 1.65e5T + 8.58e9T^{2} \)
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

−14.08264162140998563000408282294, −13.33264035578868908634176586580, −11.96057904813145789023340408521, −11.44657230668083340112240698585, −8.788928585685443676952911916437, −7.56053663229898361942920141714, −6.91581558634555231880520277382, −4.73761932669662779602835475335, −3.29962693502789694151321434494, −0.76154364176572218753187666071, 3.35177886952115890307566984523, 4.18491850096877613688237322586, 5.05004382511845517783943195156, 8.116088210581519164965973873330, 8.625163143129392673570587849666, 10.50627400217049890525185787684, 11.71608139763792213682908538468, 12.40974700404112116222600064979, 14.31574711838116275673161679754, 14.88314629322674606889324230956

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