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Properties

Label 2-810-45.29-c2-0-19
Degree $2$
Conductor $810$
Sign $0.867 + 0.497i$
Analytic cond. $22.0709$
Root an. cond. $4.69796$
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  + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (−4.52 + 2.12i)5-s + (−1.70 − 0.982i)7-s + 2.82·8-s + (0.594 − 7.04i)10-s + (3.30 + 1.90i)11-s + (−16.1 + 9.30i)13-s + (2.40 − 1.38i)14-s + (−2.00 + 3.46i)16-s + 17.6·17-s − 26.0·19-s + (8.20 + 5.71i)20-s + (−4.67 + 2.69i)22-s + (4.82 + 8.34i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.905 + 0.425i)5-s + (−0.243 − 0.140i)7-s + 0.353·8-s + (0.0594 − 0.704i)10-s + (0.300 + 0.173i)11-s + (−1.24 + 0.716i)13-s + (0.171 − 0.0992i)14-s + (−0.125 + 0.216i)16-s + 1.03·17-s − 1.37·19-s + (0.410 + 0.285i)20-s + (−0.212 + 0.122i)22-s + (0.209 + 0.363i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.867 + 0.497i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ 0.867 + 0.497i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6670866992\)
\(L(\frac12)\) \(\approx\) \(0.6670866992\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 \)
5 \( 1 + (4.52 - 2.12i)T \)
good7 \( 1 + (1.70 + 0.982i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.30 - 1.90i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (16.1 - 9.30i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 17.6T + 289T^{2} \)
19 \( 1 + 26.0T + 361T^{2} \)
23 \( 1 + (-4.82 - 8.34i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (5.20 + 3.00i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (12.6 + 21.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 39.9iT - 1.36e3T^{2} \)
41 \( 1 + (30.7 - 17.7i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-52.8 - 30.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-35.4 + 61.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 83.3T + 2.80e3T^{2} \)
59 \( 1 + (-32.3 + 18.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-2.37 + 4.11i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-8.03 + 4.63i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 34.1iT - 5.04e3T^{2} \)
73 \( 1 + 22.9iT - 5.32e3T^{2} \)
79 \( 1 + (-49.5 + 85.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-18.9 + 32.8i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 43.0iT - 7.92e3T^{2} \)
97 \( 1 + (-23.7 - 13.7i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.902726761845990706497346306703, −9.080463903956111030944826454764, −8.133481815008411547529113908096, −7.30394618942612252824163357879, −6.83102567541735682982418058184, −5.70112171777685904401781465329, −4.52686085310926065351555851758, −3.70108425251187436225647603275, −2.21343313086716392969742998167, −0.33539859705194805087604235845, 0.911271519644240798246599645848, 2.53314385776527085979808355597, 3.56733833466214506245744425620, 4.53985508801456929073252207895, 5.52468284824151915252777678934, 6.91653115001827413063995998963, 7.74643299311155859686973624693, 8.499320207732520376484977173518, 9.256627841554178852487562063608, 10.22663222876789924027333707969

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