Properties

Label 2-7-1.1-c5-0-2
Degree $2$
Conductor $7$
Sign $-1$
Analytic cond. $1.12268$
Root an. cond. $1.05956$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·2-s − 14·3-s + 68·4-s − 56·5-s + 140·6-s − 49·7-s − 360·8-s − 47·9-s + 560·10-s + 232·11-s − 952·12-s − 140·13-s + 490·14-s + 784·15-s + 1.42e3·16-s − 1.72e3·17-s + 470·18-s − 98·19-s − 3.80e3·20-s + 686·21-s − 2.32e3·22-s + 1.82e3·23-s + 5.04e3·24-s + 11·25-s + 1.40e3·26-s + 4.06e3·27-s − 3.33e3·28-s + ⋯
L(s)  = 1  − 1.76·2-s − 0.898·3-s + 17/8·4-s − 1.00·5-s + 1.58·6-s − 0.377·7-s − 1.98·8-s − 0.193·9-s + 1.77·10-s + 0.578·11-s − 1.90·12-s − 0.229·13-s + 0.668·14-s + 0.899·15-s + 1.39·16-s − 1.44·17-s + 0.341·18-s − 0.0622·19-s − 2.12·20-s + 0.339·21-s − 1.02·22-s + 0.718·23-s + 1.78·24-s + 0.00351·25-s + 0.406·26-s + 1.07·27-s − 0.803·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $-1$
Analytic conductor: \(1.12268\)
Root analytic conductor: \(1.05956\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + p^{2} T \)
good2 \( 1 + 5 p T + p^{5} T^{2} \)
3 \( 1 + 14 T + p^{5} T^{2} \)
5 \( 1 + 56 T + p^{5} T^{2} \)
11 \( 1 - 232 T + p^{5} T^{2} \)
13 \( 1 + 140 T + p^{5} T^{2} \)
17 \( 1 + 1722 T + p^{5} T^{2} \)
19 \( 1 + 98 T + p^{5} T^{2} \)
23 \( 1 - 1824 T + p^{5} T^{2} \)
29 \( 1 - 3418 T + p^{5} T^{2} \)
31 \( 1 + 7644 T + p^{5} T^{2} \)
37 \( 1 + 10398 T + p^{5} T^{2} \)
41 \( 1 + 17962 T + p^{5} T^{2} \)
43 \( 1 - 10880 T + p^{5} T^{2} \)
47 \( 1 - 9324 T + p^{5} T^{2} \)
53 \( 1 - 2262 T + p^{5} T^{2} \)
59 \( 1 + 2730 T + p^{5} T^{2} \)
61 \( 1 - 25648 T + p^{5} T^{2} \)
67 \( 1 + 48404 T + p^{5} T^{2} \)
71 \( 1 + 58560 T + p^{5} T^{2} \)
73 \( 1 - 68082 T + p^{5} T^{2} \)
79 \( 1 - 31784 T + p^{5} T^{2} \)
83 \( 1 + 20538 T + p^{5} T^{2} \)
89 \( 1 + 50582 T + p^{5} T^{2} \)
97 \( 1 + 58506 T + p^{5} T^{2} \)
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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

−20.10944672189894292223090579787, −19.23763413586165773814693661993, −17.76219309428767206123945281618, −16.72394002766407222918827889205, −15.54789416406253293357569816158, −11.88782882790828109676840958835, −10.80888703346046802716236202761, −8.815083052448570885035901859911, −6.89419253173976726633647848752, 0, 6.89419253173976726633647848752, 8.815083052448570885035901859911, 10.80888703346046802716236202761, 11.88782882790828109676840958835, 15.54789416406253293357569816158, 16.72394002766407222918827889205, 17.76219309428767206123945281618, 19.23763413586165773814693661993, 20.10944672189894292223090579787

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