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Properties

Label 2-1470-35.4-c1-0-15
Degree $2$
Conductor $1470$
Sign $0.691 - 0.722i$
Analytic cond. $11.7380$
Root an. cond. $3.42607$
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  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.0918 + 2.23i)5-s + 0.999·6-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (1.03 + 1.98i)10-s + (−0.489 + 0.847i)11-s + (0.866 − 0.499i)12-s − 0.435i·13-s + (−1.19 + 1.88i)15-s + (−0.5 − 0.866i)16-s + (2.42 + 1.39i)17-s + (0.866 + 0.499i)18-s + (3.67 + 6.35i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.0410 + 0.999i)5-s + 0.408·6-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.328 + 0.626i)10-s + (−0.147 + 0.255i)11-s + (0.249 − 0.144i)12-s − 0.120i·13-s + (−0.308 + 0.487i)15-s + (−0.125 − 0.216i)16-s + (0.587 + 0.339i)17-s + (0.204 + 0.117i)18-s + (0.842 + 1.45i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.822376153\)
\(L(\frac12)\) \(\approx\) \(2.822376153\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.0918 - 2.23i)T \)
7 \( 1 \)
good11 \( 1 + (0.489 - 0.847i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.435iT - 13T^{2} \)
17 \( 1 + (-2.42 - 1.39i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.67 - 6.35i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.89 + 1.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.74T + 29T^{2} \)
31 \( 1 + (2.48 - 4.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.01 + 2.31i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.94T + 41T^{2} \)
43 \( 1 - 9.97iT - 43T^{2} \)
47 \( 1 + (-3.81 + 2.20i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.569 - 0.328i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.13 + 7.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.66 - 9.81i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.04 + 1.18i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + (13.3 + 7.69i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.44 + 2.50i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.3iT - 83T^{2} \)
89 \( 1 + (-5.06 - 8.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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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

−9.909445112919737488257671996270, −8.947067094332328654709046524392, −7.76364022841611838576226391691, −7.32656495892307517634213525462, −6.16470267401021192443868153850, −5.49022109071871170861049383172, −4.30924917861588909262492739569, −3.46140713863318655243523895611, −2.80510543916964010108623604204, −1.61670258886776178417022659195, 0.914676589363345321904523506524, 2.37081976179870489222367107170, 3.41563856442047450061566087144, 4.39804789935006969953653982824, 5.26117641256371164649635067311, 5.92843584054695277482994164510, 7.25064454153731649087343662535, 7.54268012990217548941978107003, 8.668166757977670581560521522825, 9.122796997716213538133303108870

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