Properties

Label 2-722-1.1-c5-0-107
Degree $2$
Conductor $722$
Sign $1$
Analytic cond. $115.797$
Root an. cond. $10.7609$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 26.3·3-s + 16·4-s + 84.5·5-s + 105.·6-s + 15.2·7-s + 64·8-s + 452.·9-s + 338.·10-s − 450.·11-s + 421.·12-s + 802.·13-s + 61.0·14-s + 2.22e3·15-s + 256·16-s − 1.33e3·17-s + 1.80e3·18-s + 1.35e3·20-s + 402.·21-s − 1.80e3·22-s + 379.·23-s + 1.68e3·24-s + 4.01e3·25-s + 3.20e3·26-s + 5.51e3·27-s + 244.·28-s + 8.33e3·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.69·3-s + 0.5·4-s + 1.51·5-s + 1.19·6-s + 0.117·7-s + 0.353·8-s + 1.86·9-s + 1.06·10-s − 1.12·11-s + 0.845·12-s + 1.31·13-s + 0.0832·14-s + 2.55·15-s + 0.250·16-s − 1.11·17-s + 1.31·18-s + 0.755·20-s + 0.199·21-s − 0.793·22-s + 0.149·23-s + 0.597·24-s + 1.28·25-s + 0.931·26-s + 1.45·27-s + 0.0588·28-s + 1.83·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(115.797\)
Root analytic conductor: \(10.7609\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(9.607592082\)
\(L(\frac12)\) \(\approx\) \(9.607592082\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4T \)
19 \( 1 \)
good3 \( 1 - 26.3T + 243T^{2} \)
5 \( 1 - 84.5T + 3.12e3T^{2} \)
7 \( 1 - 15.2T + 1.68e4T^{2} \)
11 \( 1 + 450.T + 1.61e5T^{2} \)
13 \( 1 - 802.T + 3.71e5T^{2} \)
17 \( 1 + 1.33e3T + 1.41e6T^{2} \)
23 \( 1 - 379.T + 6.43e6T^{2} \)
29 \( 1 - 8.33e3T + 2.05e7T^{2} \)
31 \( 1 + 6.68e3T + 2.86e7T^{2} \)
37 \( 1 + 4.85e3T + 6.93e7T^{2} \)
41 \( 1 - 1.25e4T + 1.15e8T^{2} \)
43 \( 1 - 1.47e4T + 1.47e8T^{2} \)
47 \( 1 + 1.23e4T + 2.29e8T^{2} \)
53 \( 1 - 1.64e3T + 4.18e8T^{2} \)
59 \( 1 - 5.53e3T + 7.14e8T^{2} \)
61 \( 1 - 3.36e4T + 8.44e8T^{2} \)
67 \( 1 - 1.07e3T + 1.35e9T^{2} \)
71 \( 1 - 2.46e4T + 1.80e9T^{2} \)
73 \( 1 + 4.97e4T + 2.07e9T^{2} \)
79 \( 1 - 3.07e4T + 3.07e9T^{2} \)
83 \( 1 - 3.76e4T + 3.93e9T^{2} \)
89 \( 1 + 9.27e4T + 5.58e9T^{2} \)
97 \( 1 + 1.14e5T + 8.58e9T^{2} \)
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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

−9.531718979086245754924374271996, −8.762223308045937914511505529998, −8.096412785351007402345333202402, −6.95823405770191628138086417266, −6.08319327362038267403582419855, −5.07270450564248158859353218081, −3.97005799032390088207366239480, −2.85069774268311497839523909227, −2.28980472269412897695903782305, −1.38519175113141271517070428528, 1.38519175113141271517070428528, 2.28980472269412897695903782305, 2.85069774268311497839523909227, 3.97005799032390088207366239480, 5.07270450564248158859353218081, 6.08319327362038267403582419855, 6.95823405770191628138086417266, 8.096412785351007402345333202402, 8.762223308045937914511505529998, 9.531718979086245754924374271996

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