Properties

Label 2-7e2-49.11-c5-0-4
Degree $2$
Conductor $49$
Sign $-0.0764 - 0.997i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.71 − 3.44i)2-s + (−4.83 + 0.729i)3-s + (−0.473 + 6.32i)4-s + (−12.3 + 31.5i)5-s + (−15.4 + 19.3i)6-s + (−128. + 16.6i)7-s + (121. + 151. i)8-s + (−209. + 64.5i)9-s + (62.6 + 159. i)10-s + (−38.8 − 11.9i)11-s + (−2.31 − 30.9i)12-s + (172. + 756. i)13-s + (−419. + 504. i)14-s + (36.8 − 161. i)15-s + (772. + 116. i)16-s + (856. − 583. i)17-s + ⋯
L(s)  = 1  + (0.656 − 0.609i)2-s + (−0.310 + 0.0467i)3-s + (−0.0148 + 0.197i)4-s + (−0.221 + 0.564i)5-s + (−0.175 + 0.219i)6-s + (−0.991 + 0.128i)7-s + (0.668 + 0.838i)8-s + (−0.861 + 0.265i)9-s + (0.198 + 0.505i)10-s + (−0.0966 − 0.0298i)11-s + (−0.00464 − 0.0620i)12-s + (0.283 + 1.24i)13-s + (−0.572 + 0.688i)14-s + (0.0423 − 0.185i)15-s + (0.754 + 0.113i)16-s + (0.718 − 0.489i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0764 - 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0764 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.0764 - 0.997i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -0.0764 - 0.997i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.852378 + 0.920228i\)
\(L(\frac12)\) \(\approx\) \(0.852378 + 0.920228i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (128. - 16.6i)T \)
good2 \( 1 + (-3.71 + 3.44i)T + (2.39 - 31.9i)T^{2} \)
3 \( 1 + (4.83 - 0.729i)T + (232. - 71.6i)T^{2} \)
5 \( 1 + (12.3 - 31.5i)T + (-2.29e3 - 2.12e3i)T^{2} \)
11 \( 1 + (38.8 + 11.9i)T + (1.33e5 + 9.07e4i)T^{2} \)
13 \( 1 + (-172. - 756. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (-856. + 583. i)T + (5.18e5 - 1.32e6i)T^{2} \)
19 \( 1 + (-108. - 187. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (2.42e3 + 1.65e3i)T + (2.35e6 + 5.99e6i)T^{2} \)
29 \( 1 + (4.26e3 + 2.05e3i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (3.39e3 - 5.88e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (657. + 8.77e3i)T + (-6.85e7 + 1.03e7i)T^{2} \)
41 \( 1 + (-7.74e3 - 9.71e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (-4.00e3 + 5.01e3i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (-1.75e4 + 1.63e4i)T + (1.71e7 - 2.28e8i)T^{2} \)
53 \( 1 + (1.89e3 - 2.52e4i)T + (-4.13e8 - 6.23e7i)T^{2} \)
59 \( 1 + (-8.97e3 - 2.28e4i)T + (-5.24e8 + 4.86e8i)T^{2} \)
61 \( 1 + (-2.44e3 - 3.26e4i)T + (-8.35e8 + 1.25e8i)T^{2} \)
67 \( 1 + (2.76e4 - 4.78e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-2.88e4 + 1.38e4i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (-2.38e4 - 2.20e4i)T + (1.54e8 + 2.06e9i)T^{2} \)
79 \( 1 + (8.48e3 + 1.46e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (8.90e3 - 3.89e4i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (-5.02e4 + 1.54e4i)T + (4.61e9 - 3.14e9i)T^{2} \)
97 \( 1 + 8.24e4T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.49956757507105210811515704269, −13.70640265951144169848964010405, −12.41813554268969598174562531265, −11.58558693597139601313320448408, −10.57298039416860035689087468872, −8.929388571371175044094518872369, −7.26534272853645542444145047857, −5.70746849896536984255330229198, −3.89850658153400660875492762880, −2.61661025515008994686189839864, 0.53503488859754440142390234187, 3.60556044420101872537781617852, 5.41044282457435326255591295814, 6.20517683242748584532685169352, 7.83837208660993176857144821787, 9.495341006673842057039425595537, 10.73398917009351649989385429047, 12.38357428035788732447031192703, 13.15785671596999278705138943637, 14.36895941533327776449325106023

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