Properties

Label 2-7-1.1-c3-0-0
Degree $2$
Conductor $7$
Sign $1$
Analytic cond. $0.413013$
Root an. cond. $0.642661$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 7·4-s + 16·5-s + 2·6-s − 7·7-s + 15·8-s − 23·9-s − 16·10-s − 8·11-s + 14·12-s + 28·13-s + 7·14-s − 32·15-s + 41·16-s + 54·17-s + 23·18-s − 110·19-s − 112·20-s + 14·21-s + 8·22-s + 48·23-s − 30·24-s + 131·25-s − 28·26-s + 100·27-s + 49·28-s + ⋯
L(s)  = 1  − 0.353·2-s − 0.384·3-s − 7/8·4-s + 1.43·5-s + 0.136·6-s − 0.377·7-s + 0.662·8-s − 0.851·9-s − 0.505·10-s − 0.219·11-s + 0.336·12-s + 0.597·13-s + 0.133·14-s − 0.550·15-s + 0.640·16-s + 0.770·17-s + 0.301·18-s − 1.32·19-s − 1.25·20-s + 0.145·21-s + 0.0775·22-s + 0.435·23-s − 0.255·24-s + 1.04·25-s − 0.211·26-s + 0.712·27-s + 0.330·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $1$
Analytic conductor: \(0.413013\)
Root analytic conductor: \(0.642661\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5995661579\)
\(L(\frac12)\) \(\approx\) \(0.5995661579\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + p T \)
good2 \( 1 + T + p^{3} T^{2} \)
3 \( 1 + 2 T + p^{3} T^{2} \)
5 \( 1 - 16 T + p^{3} T^{2} \)
11 \( 1 + 8 T + p^{3} T^{2} \)
13 \( 1 - 28 T + p^{3} T^{2} \)
17 \( 1 - 54 T + p^{3} T^{2} \)
19 \( 1 + 110 T + p^{3} T^{2} \)
23 \( 1 - 48 T + p^{3} T^{2} \)
29 \( 1 + 110 T + p^{3} T^{2} \)
31 \( 1 - 12 T + p^{3} T^{2} \)
37 \( 1 + 246 T + p^{3} T^{2} \)
41 \( 1 - 182 T + p^{3} T^{2} \)
43 \( 1 - 128 T + p^{3} T^{2} \)
47 \( 1 - 324 T + p^{3} T^{2} \)
53 \( 1 + 162 T + p^{3} T^{2} \)
59 \( 1 - 810 T + p^{3} T^{2} \)
61 \( 1 + 8 p T + p^{3} T^{2} \)
67 \( 1 - 244 T + p^{3} T^{2} \)
71 \( 1 + 768 T + p^{3} T^{2} \)
73 \( 1 + 702 T + p^{3} T^{2} \)
79 \( 1 - 440 T + p^{3} T^{2} \)
83 \( 1 + 1302 T + p^{3} T^{2} \)
89 \( 1 - 730 T + p^{3} T^{2} \)
97 \( 1 - 294 T + p^{3} T^{2} \)
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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

−22.29829705980502549266677508752, −21.01874672820152951151166325098, −18.98691076138796378062634315761, −17.66603114438279377377088788569, −16.84418190446002259612227390359, −14.24617763309018446443187607209, −13.00459047543158583812879328378, −10.42619845921982396207673495364, −8.921565108109471505669921413512, −5.74560490678666676322759669793, 5.74560490678666676322759669793, 8.921565108109471505669921413512, 10.42619845921982396207673495364, 13.00459047543158583812879328378, 14.24617763309018446443187607209, 16.84418190446002259612227390359, 17.66603114438279377377088788569, 18.98691076138796378062634315761, 21.01874672820152951151166325098, 22.29829705980502549266677508752

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