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Properties

Label 2-7-7.4-c3-0-0
Degree $2$
Conductor $7$
Sign $0.968 + 0.250i$
Analytic cond. $0.413013$
Root an. cond. $0.642661$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−3.5 + 6.06i)3-s + (2.00 − 3.46i)4-s + (−3.5 − 6.06i)5-s + 14·6-s + (14 + 12.1i)7-s − 24·8-s + (−11 − 19.0i)9-s + (−7 + 12.1i)10-s + (2.5 − 4.33i)11-s + (13.9 + 24.2i)12-s − 14·13-s + (7 − 36.3i)14-s + 49·15-s + (8.00 + 13.8i)16-s + (10.5 − 18.1i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.673 + 1.16i)3-s + (0.250 − 0.433i)4-s + (−0.313 − 0.542i)5-s + 0.952·6-s + (0.755 + 0.654i)7-s − 1.06·8-s + (−0.407 − 0.705i)9-s + (−0.221 + 0.383i)10-s + (0.0685 − 0.118i)11-s + (0.336 + 0.583i)12-s − 0.298·13-s + (0.133 − 0.694i)14-s + 0.843·15-s + (0.125 + 0.216i)16-s + (0.149 − 0.259i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(0.413013\)
Root analytic conductor: \(0.642661\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :3/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.592926 - 0.0755587i\)
\(L(\frac12)\) \(\approx\) \(0.592926 - 0.0755587i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-14 - 12.1i)T \)
good2 \( 1 + (1 + 1.73i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (3.5 - 6.06i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (3.5 + 6.06i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 14T + 2.19e3T^{2} \)
17 \( 1 + (-10.5 + 18.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (24.5 + 42.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-79.5 - 137. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 58T + 2.43e4T^{2} \)
31 \( 1 + (73.5 - 127. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (109.5 + 189. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 350T + 6.89e4T^{2} \)
43 \( 1 + 124T + 7.95e4T^{2} \)
47 \( 1 + (262.5 + 454. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (151.5 - 262. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-52.5 + 90.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-206.5 - 357. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (207.5 - 359. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 432T + 3.57e5T^{2} \)
73 \( 1 + (-556.5 + 963. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-51.5 - 89.2i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.09e3T + 5.71e5T^{2} \)
89 \( 1 + (-164.5 - 284. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 882T + 9.12e5T^{2} \)
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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

−21.73262490716289823117921584584, −20.91822196939805874340971609810, −19.50153937985057617696689143507, −17.75884298513280071891956407611, −16.09259512717138983717111855239, −14.95994334252734381646800591924, −11.91232141937324149059092308649, −10.79042177854964714767970907634, −9.170771798623879815197307093822, −5.20239540052751869909136213505, 6.69000228021965412375783704525, 7.86320333232698952994279195127, 11.23114614585979948876648558038, 12.63773290481159472105769105934, 14.75073122826914210166577738781, 16.77242482888655925605925920086, 17.71699999773205043957589791903, 18.87238470644861754687131275082, 20.83702434468867677778925371131, 22.75658303906369405879076936982

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