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Properties

Label 2-1470-105.104-c1-0-33
Degree $2$
Conductor $1470$
Sign $0.189 - 0.981i$
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  + 2-s + (−1.29 + 1.14i)3-s + 4-s + (1.07 + 1.96i)5-s + (−1.29 + 1.14i)6-s + 8-s + (0.375 − 2.97i)9-s + (1.07 + 1.96i)10-s − 0.115i·11-s + (−1.29 + 1.14i)12-s + 5.60·13-s + (−3.64 − 1.31i)15-s + 16-s + 1.39i·17-s + (0.375 − 2.97i)18-s + 1.52i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.750 + 0.661i)3-s + 0.5·4-s + (0.479 + 0.877i)5-s + (−0.530 + 0.467i)6-s + 0.353·8-s + (0.125 − 0.992i)9-s + (0.339 + 0.620i)10-s − 0.0348i·11-s + (−0.375 + 0.330i)12-s + 1.55·13-s + (−0.940 − 0.340i)15-s + 0.250·16-s + 0.337i·17-s + (0.0884 − 0.701i)18-s + 0.349i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.398867241\)
\(L(\frac12)\) \(\approx\) \(2.398867241\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.29 - 1.14i)T \)
5 \( 1 + (-1.07 - 1.96i)T \)
7 \( 1 \)
good11 \( 1 + 0.115iT - 11T^{2} \)
13 \( 1 - 5.60T + 13T^{2} \)
17 \( 1 - 1.39iT - 17T^{2} \)
19 \( 1 - 1.52iT - 19T^{2} \)
23 \( 1 - 6.58T + 23T^{2} \)
29 \( 1 + 8.97iT - 29T^{2} \)
31 \( 1 - 7.15iT - 31T^{2} \)
37 \( 1 + 1.70iT - 37T^{2} \)
41 \( 1 + 3.82T + 41T^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 + 2.37iT - 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 + 3.32T + 59T^{2} \)
61 \( 1 + 5.12iT - 61T^{2} \)
67 \( 1 - 0.689iT - 67T^{2} \)
71 \( 1 - 6.77iT - 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 - 6.24iT - 83T^{2} \)
89 \( 1 + 6.83T + 89T^{2} \)
97 \( 1 - 16.3T + 97T^{2} \)
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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

−9.918895585308575800668097429525, −9.072743029019136096516060425467, −7.991847790422367576890384804047, −6.77409075982574148245540531247, −6.28590749429401374823852340443, −5.64784879147690410501351917895, −4.68048327165738248649082248445, −3.68453452268906557690702739403, −3.01064433277364389660455829044, −1.42514355161826600431884792428, 0.948202739767666964225165804326, 1.88957733523596298145504718883, 3.28484416380508373397669892776, 4.55337568003464628212759128921, 5.22738499041217946286564155463, 5.95028662885815488840259991247, 6.65654600912952319383448165553, 7.50774374822296191186613026843, 8.540034302893608414673993386104, 9.178450327144766579122369201350

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