Properties

Label 2-7-7.4-c5-0-0
Degree $2$
Conductor $7$
Sign $0.792 - 0.610i$
Analytic cond. $1.12268$
Root an. cond. $1.05956$
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  + (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

Degree: \(2\)
Conductor: \(7\)
Sign: $0.792 - 0.610i$
Analytic conductor: \(1.12268\)
Root analytic conductor: \(1.05956\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :5/2),\ 0.792 - 0.610i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.16538 + 0.396739i\)
\(L(\frac12)\) \(\approx\) \(1.16538 + 0.396739i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−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

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