Properties

Label 2-1710-1.1-c3-0-2
Degree $2$
Conductor $1710$
Sign $1$
Analytic cond. $100.893$
Root an. cond. $10.0445$
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  − 2·2-s + 4·4-s + 5·5-s − 24·7-s − 8·8-s − 10·10-s − 32·11-s + 2·13-s + 48·14-s + 16·16-s − 106·17-s − 19·19-s + 20·20-s + 64·22-s − 152·23-s + 25·25-s − 4·26-s − 96·28-s − 90·29-s + 52·31-s − 32·32-s + 212·34-s − 120·35-s + 306·37-s + 38·38-s − 40·40-s − 62·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.29·7-s − 0.353·8-s − 0.316·10-s − 0.877·11-s + 0.0426·13-s + 0.916·14-s + 1/4·16-s − 1.51·17-s − 0.229·19-s + 0.223·20-s + 0.620·22-s − 1.37·23-s + 1/5·25-s − 0.0301·26-s − 0.647·28-s − 0.576·29-s + 0.301·31-s − 0.176·32-s + 1.06·34-s − 0.579·35-s + 1.35·37-s + 0.162·38-s − 0.158·40-s − 0.236·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(100.893\)
Root analytic conductor: \(10.0445\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5273046789\)
\(L(\frac12)\) \(\approx\) \(0.5273046789\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 \)
5 \( 1 - p T \)
19 \( 1 + p T \)
good7 \( 1 + 24 T + p^{3} T^{2} \)
11 \( 1 + 32 T + p^{3} T^{2} \)
13 \( 1 - 2 T + p^{3} T^{2} \)
17 \( 1 + 106 T + p^{3} T^{2} \)
23 \( 1 + 152 T + p^{3} T^{2} \)
29 \( 1 + 90 T + p^{3} T^{2} \)
31 \( 1 - 52 T + p^{3} T^{2} \)
37 \( 1 - 306 T + p^{3} T^{2} \)
41 \( 1 + 62 T + p^{3} T^{2} \)
43 \( 1 + 268 T + p^{3} T^{2} \)
47 \( 1 + 456 T + p^{3} T^{2} \)
53 \( 1 - 6 p T + p^{3} T^{2} \)
59 \( 1 + 300 T + p^{3} T^{2} \)
61 \( 1 - 502 T + p^{3} T^{2} \)
67 \( 1 + 644 T + p^{3} T^{2} \)
71 \( 1 - 608 T + p^{3} T^{2} \)
73 \( 1 + 198 T + p^{3} T^{2} \)
79 \( 1 - 260 T + p^{3} T^{2} \)
83 \( 1 - 1248 T + p^{3} T^{2} \)
89 \( 1 + 110 T + p^{3} T^{2} \)
97 \( 1 + 574 T + p^{3} T^{2} \)
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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

−9.052274315484558281870937635480, −8.291101169711890054004272740922, −7.44038431303817979881594402154, −6.43305625582112608754777824505, −6.15571774514570208788075462851, −4.95224949597489704741659247441, −3.76926218380155516096984227106, −2.70569402617748958044089463171, −1.96059652869516783373282177230, −0.35902937402663873825912277815, 0.35902937402663873825912277815, 1.96059652869516783373282177230, 2.70569402617748958044089463171, 3.76926218380155516096984227106, 4.95224949597489704741659247441, 6.15571774514570208788075462851, 6.43305625582112608754777824505, 7.44038431303817979881594402154, 8.291101169711890054004272740922, 9.052274315484558281870937635480

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