Properties

Label 2-7e2-49.2-c5-0-2
Degree $2$
Conductor $49$
Sign $0.750 - 0.660i$
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  + (−10.0 − 3.09i)2-s + (9.56 − 24.3i)3-s + (64.7 + 44.1i)4-s + (−25.7 + 3.88i)5-s + (−171. + 214. i)6-s + (24.4 + 127. i)7-s + (−303. − 381. i)8-s + (−323. − 300. i)9-s + (270. + 40.8i)10-s + (−479. + 444. i)11-s + (1.69e3 − 1.15e3i)12-s + (195. + 855. i)13-s + (148. − 1.35e3i)14-s + (−151. + 665. i)15-s + (954. + 2.43e3i)16-s + (59.6 + 795. i)17-s + ⋯
L(s)  = 1  + (−1.77 − 0.547i)2-s + (0.613 − 1.56i)3-s + (2.02 + 1.37i)4-s + (−0.461 + 0.0695i)5-s + (−1.94 + 2.43i)6-s + (0.188 + 0.981i)7-s + (−1.67 − 2.10i)8-s + (−1.33 − 1.23i)9-s + (0.856 + 0.129i)10-s + (−1.19 + 1.10i)11-s + (3.39 − 2.31i)12-s + (0.320 + 1.40i)13-s + (0.202 − 1.84i)14-s + (−0.174 + 0.763i)15-s + (0.932 + 2.37i)16-s + (0.0500 + 0.667i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.750 - 0.660i$
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: \((2,\ 49,\ (\ :5/2),\ 0.750 - 0.660i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.375858 + 0.141907i\)
\(L(\frac12)\) \(\approx\) \(0.375858 + 0.141907i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-24.4 - 127. i)T \)
good2 \( 1 + (10.0 + 3.09i)T + (26.4 + 18.0i)T^{2} \)
3 \( 1 + (-9.56 + 24.3i)T + (-178. - 165. i)T^{2} \)
5 \( 1 + (25.7 - 3.88i)T + (2.98e3 - 921. i)T^{2} \)
11 \( 1 + (479. - 444. i)T + (1.20e4 - 1.60e5i)T^{2} \)
13 \( 1 + (-195. - 855. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (-59.6 - 795. i)T + (-1.40e6 + 2.11e5i)T^{2} \)
19 \( 1 + (-702. + 1.21e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-68.6 + 916. i)T + (-6.36e6 - 9.59e5i)T^{2} \)
29 \( 1 + (675. + 325. i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (-18.1 - 31.3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-132. + 90.4i)T + (2.53e7 - 6.45e7i)T^{2} \)
41 \( 1 + (-5.24e3 - 6.57e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (6.76e3 - 8.48e3i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (1.24e4 + 3.83e3i)T + (1.89e8 + 1.29e8i)T^{2} \)
53 \( 1 + (1.22e4 + 8.37e3i)T + (1.52e8 + 3.89e8i)T^{2} \)
59 \( 1 + (-2.58e4 - 3.89e3i)T + (6.83e8 + 2.10e8i)T^{2} \)
61 \( 1 + (2.29e4 - 1.56e4i)T + (3.08e8 - 7.86e8i)T^{2} \)
67 \( 1 + (-3.00e4 - 5.20e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (1.70e4 - 8.21e3i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (2.02e3 - 624. i)T + (1.71e9 - 1.16e9i)T^{2} \)
79 \( 1 + (-3.91e4 + 6.77e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (1.72e4 - 7.55e4i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (-3.37e4 - 3.13e4i)T + (4.17e8 + 5.56e9i)T^{2} \)
97 \( 1 + 9.73e4T + 8.58e9T^{2} \)
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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

−14.94310704366504725867209332754, −13.09322546957983288006731407001, −12.11781537729962330689906616827, −11.33562143929035626912843812881, −9.572030135675138606931131843637, −8.477156765634679954102608857971, −7.70457689410854960316950852240, −6.70209698434516896270569473196, −2.57859113725447863802435519071, −1.66399818998068330966598013923, 0.33005915453094066532964352491, 3.28144114889075762354747585598, 5.47128578797054223007444508553, 7.77961443027511085894103209013, 8.272326236461238606864526072863, 9.666681012579656374484098612421, 10.48700568846243969189668273114, 11.09928538019523300811651963753, 13.85394167494004377441141809112, 15.17999885518821810982097300666

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