# Properties

 Degree 2 Conductor 7 Sign $0.792 - 0.610i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (2.54 + 4.40i)2-s + (−1.04 + 1.80i)3-s + (3.08 − 5.33i)4-s + (−20.9 − 36.2i)5-s − 10.5·6-s + (−127. + 25.2i)7-s + 193.·8-s + (119. + 206. i)9-s + (106. − 184. i)10-s + (−36.0 + 62.4i)11-s + (6.42 + 11.1i)12-s − 632.·13-s + (−434. − 495. i)14-s + 87.1·15-s + (394. + 683. i)16-s + (987. − 1.71e3i)17-s + ⋯
 L(s)  = 1 + (0.449 + 0.778i)2-s + (−0.0668 + 0.115i)3-s + (0.0963 − 0.166i)4-s + (−0.374 − 0.647i)5-s − 0.120·6-s + (−0.980 + 0.194i)7-s + 1.07·8-s + (0.491 + 0.850i)9-s + (0.336 − 0.582i)10-s + (−0.0898 + 0.155i)11-s + (0.0128 + 0.0222i)12-s − 1.03·13-s + (−0.592 − 0.675i)14-s + 0.0999·15-s + (0.385 + 0.667i)16-s + (0.829 − 1.43i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$7$$ $$\varepsilon$$ = $0.792 - 0.610i$ motivic weight = $$5$$ character : $\chi_{7} (4, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 7,\ (\ :5/2),\ 0.792 - 0.610i)$ $L(3)$ $\approx$ $1.16538 + 0.396739i$ $L(\frac12)$ $\approx$ $1.16538 + 0.396739i$ $L(\frac{7}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 7$, $$F_p$$ is a polynomial of degree 2. If $p = 7$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 $$1 + (127. - 25.2i)T$$
good2 $$1 + (-2.54 - 4.40i)T + (-16 + 27.7i)T^{2}$$
3 $$1 + (1.04 - 1.80i)T + (-121.5 - 210. i)T^{2}$$
5 $$1 + (20.9 + 36.2i)T + (-1.56e3 + 2.70e3i)T^{2}$$
11 $$1 + (36.0 - 62.4i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + 632.T + 3.71e5T^{2}$$
17 $$1 + (-987. + 1.71e3i)T + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (-932. - 1.61e3i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (206. + 358. i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 - 731.T + 2.05e7T^{2}$$
31 $$1 + (3.06e3 - 5.30e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (5.17e3 + 8.96e3i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 + 3.52e3T + 1.15e8T^{2}$$
43 $$1 + 1.45e4T + 1.47e8T^{2}$$
47 $$1 + (-1.07e4 - 1.85e4i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + (6.28e3 - 1.08e4i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (-1.80e4 + 3.12e4i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (2.01e3 + 3.48e3i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (7.78e3 - 1.34e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 - 1.21e4T + 1.80e9T^{2}$$
73 $$1 + (9.79e3 - 1.69e4i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (1.80e4 + 3.12e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + 2.45e4T + 3.93e9T^{2}$$
89 $$1 + (-3.51e4 - 6.08e4i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 - 1.05e5T + 8.58e9T^{2}$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−22.17624364857314182956509306672, −20.17980668027821707748323859189, −18.96717658600509364384564236085, −16.47192746263695183583764465299, −15.94156673157555611824397647322, −14.13425557388889850963185316499, −12.44501284768504795812830520917, −9.998802688611086109464520430262, −7.37732792484842868284117396695, −5.12553753782728504599967213955, 3.43551399195511712236903657816, 7.10655360964114683232862426342, 10.14284762309064775210492991809, 11.90327030397803109767832616100, 13.11981359757681676277581925418, 15.15826741594825591264719986097, 16.93141700154744777014958807001, 18.92289127552869610966216251665, 20.00320751269634992621199298293, 21.57308437615047078910039825640