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Properties

Label 2-1710-95.37-c1-0-25
Degree $2$
Conductor $1710$
Sign $0.999 + 0.0108i$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (1.29 − 1.81i)5-s + (−0.728 − 0.728i)7-s + (0.707 + 0.707i)8-s + (0.367 + 2.20i)10-s + 4.80·11-s + (0.531 + 0.531i)13-s + 1.03·14-s − 1.00·16-s + (3.72 + 3.72i)17-s + (−2.90 + 3.24i)19-s + (−1.81 − 1.29i)20-s + (−3.39 + 3.39i)22-s + (4.07 − 4.07i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.581 − 0.813i)5-s + (−0.275 − 0.275i)7-s + (0.250 + 0.250i)8-s + (0.116 + 0.697i)10-s + 1.44·11-s + (0.147 + 0.147i)13-s + 0.275·14-s − 0.250·16-s + (0.904 + 0.904i)17-s + (−0.666 + 0.745i)19-s + (−0.406 − 0.290i)20-s + (−0.724 + 0.724i)22-s + (0.850 − 0.850i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.999 + 0.0108i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.999 + 0.0108i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.619320066\)
\(L(\frac12)\) \(\approx\) \(1.619320066\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-1.29 + 1.81i)T \)
19 \( 1 + (2.90 - 3.24i)T \)
good7 \( 1 + (0.728 + 0.728i)T + 7iT^{2} \)
11 \( 1 - 4.80T + 11T^{2} \)
13 \( 1 + (-0.531 - 0.531i)T + 13iT^{2} \)
17 \( 1 + (-3.72 - 3.72i)T + 17iT^{2} \)
23 \( 1 + (-4.07 + 4.07i)T - 23iT^{2} \)
29 \( 1 + 0.494T + 29T^{2} \)
31 \( 1 - 8.62iT - 31T^{2} \)
37 \( 1 + (-5.47 + 5.47i)T - 37iT^{2} \)
41 \( 1 - 5.82iT - 41T^{2} \)
43 \( 1 + (3.04 - 3.04i)T - 43iT^{2} \)
47 \( 1 + (0.910 + 0.910i)T + 47iT^{2} \)
53 \( 1 + (3.53 + 3.53i)T + 53iT^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 + 5.32T + 61T^{2} \)
67 \( 1 + (-9.19 + 9.19i)T - 67iT^{2} \)
71 \( 1 - 2.06iT - 71T^{2} \)
73 \( 1 + (-3.31 + 3.31i)T - 73iT^{2} \)
79 \( 1 - 2.77T + 79T^{2} \)
83 \( 1 + (-3.57 + 3.57i)T - 83iT^{2} \)
89 \( 1 + 1.08T + 89T^{2} \)
97 \( 1 + (-2.81 + 2.81i)T - 97iT^{2} \)
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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

−9.222480795552350618530907158440, −8.591366778574886238421521237852, −7.964899453506586300752686792596, −6.68884235252683992299885598195, −6.35806009483632323443197295231, −5.40861254983769501177888592226, −4.44691317456442945796125222637, −3.52613955909876067951569906747, −1.83314722742790328978543828152, −0.977503926332904152289526480557, 1.05747703714534562030669056503, 2.31296813419254041203079018284, 3.16136223009154099546074478748, 4.04665317099066825895617323034, 5.35151818565035426509975927084, 6.29828217988764738197758631732, 6.94855384685551550203263471289, 7.72849704369886793447882753209, 8.862924068740537769042621048627, 9.489661186141455592140402271750

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