Properties

Label 2-7e2-49.11-c5-0-17
Degree $2$
Conductor $49$
Sign $-0.998 + 0.0509i$
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  + (1.26 − 1.17i)2-s + (−10.7 + 1.61i)3-s + (−2.17 + 28.9i)4-s + (36.0 − 91.7i)5-s + (−11.6 + 14.5i)6-s + (−80.4 + 101. i)7-s + (65.4 + 82.1i)8-s + (−119. + 36.9i)9-s + (−61.9 − 157. i)10-s + (−649. − 200. i)11-s + (−23.5 − 314. i)12-s + (−169. − 740. i)13-s + (17.5 + 222. i)14-s + (−237. + 1.04e3i)15-s + (−740. − 111. i)16-s + (−1.24e3 + 849. i)17-s + ⋯
L(s)  = 1  + (0.222 − 0.206i)2-s + (−0.688 + 0.103i)3-s + (−0.0678 + 0.904i)4-s + (0.644 − 1.64i)5-s + (−0.131 + 0.165i)6-s + (−0.620 + 0.784i)7-s + (0.361 + 0.453i)8-s + (−0.492 + 0.151i)9-s + (−0.195 − 0.499i)10-s + (−1.61 − 0.499i)11-s + (−0.0472 − 0.629i)12-s + (−0.277 − 1.21i)13-s + (0.0239 + 0.303i)14-s + (−0.273 + 1.19i)15-s + (−0.722 − 0.108i)16-s + (−1.04 + 0.712i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0509i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.998 + 0.0509i$
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: \((2,\ 49,\ (\ :5/2),\ -0.998 + 0.0509i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.00336763 - 0.132036i\)
\(L(\frac12)\) \(\approx\) \(0.00336763 - 0.132036i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (80.4 - 101. i)T \)
good2 \( 1 + (-1.26 + 1.17i)T + (2.39 - 31.9i)T^{2} \)
3 \( 1 + (10.7 - 1.61i)T + (232. - 71.6i)T^{2} \)
5 \( 1 + (-36.0 + 91.7i)T + (-2.29e3 - 2.12e3i)T^{2} \)
11 \( 1 + (649. + 200. i)T + (1.33e5 + 9.07e4i)T^{2} \)
13 \( 1 + (169. + 740. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (1.24e3 - 849. i)T + (5.18e5 - 1.32e6i)T^{2} \)
19 \( 1 + (-57.9 - 100. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-2.65e3 - 1.80e3i)T + (2.35e6 + 5.99e6i)T^{2} \)
29 \( 1 + (-1.70e3 - 823. i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (-560. + 971. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (231. + 3.08e3i)T + (-6.85e7 + 1.03e7i)T^{2} \)
41 \( 1 + (3.43e3 + 4.31e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (-2.24e3 + 2.81e3i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (1.16e4 - 1.07e4i)T + (1.71e7 - 2.28e8i)T^{2} \)
53 \( 1 + (94.2 - 1.25e3i)T + (-4.13e8 - 6.23e7i)T^{2} \)
59 \( 1 + (-9.90e3 - 2.52e4i)T + (-5.24e8 + 4.86e8i)T^{2} \)
61 \( 1 + (-1.90e3 - 2.54e4i)T + (-8.35e8 + 1.25e8i)T^{2} \)
67 \( 1 + (-5.05e3 + 8.76e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (7.04e3 - 3.39e3i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (2.26e4 + 2.10e4i)T + (1.54e8 + 2.06e9i)T^{2} \)
79 \( 1 + (7.02e3 + 1.21e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-9.86e3 + 4.32e4i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (-6.62e4 + 2.04e4i)T + (4.61e9 - 3.14e9i)T^{2} \)
97 \( 1 + 9.16e4T + 8.58e9T^{2} \)
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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

−13.17869177526792547372112219625, −13.06996063124572562929921979235, −12.01307140035365008291250317769, −10.61199531872043769608791711554, −8.949306568740323123646442841938, −8.135465907823317153346709118467, −5.70799145947819450107600540718, −4.99945888075291111531972721690, −2.70976853462368353344519269385, −0.06288240263439529291404349451, 2.57812326945923427683496884826, 4.98808882050129565853608264660, 6.52146562490609643940809181353, 6.92467484108780245937835941400, 9.650120736278017225310760170451, 10.55600751699799098897680859585, 11.22780810313542876990376465956, 13.23984851587485510653772054989, 14.06198619868563718728369419760, 14.99471302698557673616050072180

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