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Properties

Label 2-1740-29.17-c2-0-22
Degree $2$
Conductor $1740$
Sign $0.720 - 0.693i$
Analytic cond. $47.4115$
Root an. cond. $6.88560$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 1.22i)3-s + 2.23i·5-s + 4.92·7-s − 2.99i·9-s + (−4.50 + 4.50i)11-s + 2.37i·13-s + (−2.73 − 2.73i)15-s + (9.67 − 9.67i)17-s + (3.07 − 3.07i)19-s + (−6.02 + 6.02i)21-s + 20.3·23-s − 5.00·25-s + (3.67 + 3.67i)27-s + (−4.95 − 28.5i)29-s + (31.2 − 31.2i)31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + 0.447i·5-s + 0.702·7-s − 0.333i·9-s + (−0.409 + 0.409i)11-s + 0.183i·13-s + (−0.182 − 0.182i)15-s + (0.569 − 0.569i)17-s + (0.161 − 0.161i)19-s + (−0.286 + 0.286i)21-s + 0.885·23-s − 0.200·25-s + (0.136 + 0.136i)27-s + (−0.170 − 0.985i)29-s + (1.00 − 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $0.720 - 0.693i$
Analytic conductor: \(47.4115\)
Root analytic conductor: \(6.88560\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1740} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1740,\ (\ :1),\ 0.720 - 0.693i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.887097734\)
\(L(\frac12)\) \(\approx\) \(1.887097734\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 - 2.23iT \)
29 \( 1 + (4.95 + 28.5i)T \)
good7 \( 1 - 4.92T + 49T^{2} \)
11 \( 1 + (4.50 - 4.50i)T - 121iT^{2} \)
13 \( 1 - 2.37iT - 169T^{2} \)
17 \( 1 + (-9.67 + 9.67i)T - 289iT^{2} \)
19 \( 1 + (-3.07 + 3.07i)T - 361iT^{2} \)
23 \( 1 - 20.3T + 529T^{2} \)
31 \( 1 + (-31.2 + 31.2i)T - 961iT^{2} \)
37 \( 1 + (-5.24 - 5.24i)T + 1.36e3iT^{2} \)
41 \( 1 + (-47.2 - 47.2i)T + 1.68e3iT^{2} \)
43 \( 1 + (51.9 - 51.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (4.56 + 4.56i)T + 2.20e3iT^{2} \)
53 \( 1 - 17.4T + 2.80e3T^{2} \)
59 \( 1 - 60.6T + 3.48e3T^{2} \)
61 \( 1 + (-77.4 + 77.4i)T - 3.72e3iT^{2} \)
67 \( 1 - 24.2iT - 4.48e3T^{2} \)
71 \( 1 + 63.5iT - 5.04e3T^{2} \)
73 \( 1 + (32.9 + 32.9i)T + 5.32e3iT^{2} \)
79 \( 1 + (-24.8 + 24.8i)T - 6.24e3iT^{2} \)
83 \( 1 - 79.9T + 6.88e3T^{2} \)
89 \( 1 + (63.0 - 63.0i)T - 7.92e3iT^{2} \)
97 \( 1 + (47.9 + 47.9i)T + 9.40e3iT^{2} \)
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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

−9.590995726245984946870039526799, −8.277717178201367736287630702839, −7.71602186350652925358705906325, −6.79816511080163856551369280481, −5.97584734537615331559280638660, −5.00106058755262184016067999818, −4.46411479059993638615727040759, −3.26223059846540202122628263232, −2.27193640175100850975630608511, −0.849052439393846156991305048602, 0.75327863446040522808491353251, 1.68102380095239282873275377723, 2.95891538900941435914797183812, 4.10729107553766335825124323428, 5.25526476278162915284799781140, 5.50115528426185268786962762761, 6.71602923441597617264191430564, 7.43884468939588644810343242426, 8.363116187115591691593620802943, 8.714189049447911516546057788216

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