Properties

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

Origins

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Normalization:  

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  :  $\chi_{7} (2, \cdot )$
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

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

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