Properties

Label 2-722-1.1-c1-0-7
Degree $2$
Conductor $722$
Sign $-1$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.79·3-s + 4-s − 2.34·5-s + 2.79·6-s − 1.28·7-s − 8-s + 4.80·9-s + 2.34·10-s + 5.75·11-s − 2.79·12-s − 0.304·13-s + 1.28·14-s + 6.54·15-s + 16-s + 4.18·17-s − 4.80·18-s − 2.34·20-s + 3.58·21-s − 5.75·22-s − 6.47·23-s + 2.79·24-s + 0.497·25-s + 0.304·26-s − 5.04·27-s − 1.28·28-s + 3.12·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.61·3-s + 0.5·4-s − 1.04·5-s + 1.14·6-s − 0.485·7-s − 0.353·8-s + 1.60·9-s + 0.741·10-s + 1.73·11-s − 0.806·12-s − 0.0843·13-s + 0.343·14-s + 1.69·15-s + 0.250·16-s + 1.01·17-s − 1.13·18-s − 0.524·20-s + 0.782·21-s − 1.22·22-s − 1.35·23-s + 0.570·24-s + 0.0994·25-s + 0.0596·26-s − 0.970·27-s − 0.242·28-s + 0.580·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -1)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
19 \( 1 \)
good3 \( 1 + 2.79T + 3T^{2} \)
5 \( 1 + 2.34T + 5T^{2} \)
7 \( 1 + 1.28T + 7T^{2} \)
11 \( 1 - 5.75T + 11T^{2} \)
13 \( 1 + 0.304T + 13T^{2} \)
17 \( 1 - 4.18T + 17T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 - 3.12T + 29T^{2} \)
31 \( 1 + 6.44T + 31T^{2} \)
37 \( 1 - 3.97T + 37T^{2} \)
41 \( 1 - 5.01T + 41T^{2} \)
43 \( 1 + 0.989T + 43T^{2} \)
47 \( 1 + 4.39T + 47T^{2} \)
53 \( 1 + 3.29T + 53T^{2} \)
59 \( 1 + 3.31T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 - 4.38T + 67T^{2} \)
71 \( 1 - 4.41T + 71T^{2} \)
73 \( 1 + 2.26T + 73T^{2} \)
79 \( 1 + 8.57T + 79T^{2} \)
83 \( 1 + 9.76T + 83T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−10.00145599627902110158431853334, −9.370034991594865017601644426718, −8.125546927129838533570999708001, −7.26396626986954964448708158247, −6.42471364534263534342362214620, −5.81555434817150224211235426915, −4.43222437249828005509018297848, −3.55182529587266362830914078372, −1.32043612764107317324511906113, 0, 1.32043612764107317324511906113, 3.55182529587266362830914078372, 4.43222437249828005509018297848, 5.81555434817150224211235426915, 6.42471364534263534342362214620, 7.26396626986954964448708158247, 8.125546927129838533570999708001, 9.370034991594865017601644426718, 10.00145599627902110158431853334

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