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Properties

Label 2-1470-105.104-c1-0-32
Degree $2$
Conductor $1470$
Sign $0.997 - 0.0650i$
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 + (−0.854 − 3.77i)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 + (−0.220 − 0.975i)15-s + 0.250·16-s + 0.337i·17-s + (−0.0884 + 0.701i)18-s − 0.349i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.997 - 0.0650i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.997 - 0.0650i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7345845853\)
\(L(\frac12)\) \(\approx\) \(0.7345845853\)
\(L(\frac{3}{2})\) 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

−9.723191457000683957770761379677, −8.724460562777815784288462206372, −8.010507779293228890670213085432, −7.06277238504205187980474458956, −6.15527782598464344984655823875, −5.78706720811139028674045961844, −4.12321268938162450462099077406, −3.70518407935751383574358778426, −2.29177372752986607616407307473, −0.55620182185893995602327570099, 0.935560254443598710756920766358, 1.74885275509753650821682103268, 3.40845371090835803972247306856, 4.56784569202594654250104752699, 5.57961429107770213671953600602, 6.27507927201659238611031104869, 7.18442828811649965751196163578, 8.016539460028157824107978297716, 8.536929560178540873007291536411, 9.341021446269287585673031898861

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