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Properties

Label 2-1620-9.5-c2-0-10
Degree $2$
Conductor $1620$
Sign $0.0871 - 0.996i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
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  + (−1.93 − 1.11i)5-s + (5.78 + 10.0i)7-s + (−2.14 + 1.23i)11-s + (−2.00 + 3.46i)13-s − 13.4i·17-s + 34.7·19-s + (7.48 + 4.32i)23-s + (2.5 + 4.33i)25-s + (−17.1 + 9.89i)29-s + (0.0722 − 0.125i)31-s − 25.8i·35-s − 12.6·37-s + (−5.90 − 3.40i)41-s + (14.4 + 25.0i)43-s + (54.7 − 31.6i)47-s + ⋯
L(s)  = 1  + (−0.387 − 0.223i)5-s + (0.826 + 1.43i)7-s + (−0.194 + 0.112i)11-s + (−0.154 + 0.266i)13-s − 0.792i·17-s + 1.83·19-s + (0.325 + 0.187i)23-s + (0.100 + 0.173i)25-s + (−0.590 + 0.341i)29-s + (0.00233 − 0.00403i)31-s − 0.739i·35-s − 0.342·37-s + (−0.143 − 0.0831i)41-s + (0.336 + 0.582i)43-s + (1.16 − 0.672i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.0871 - 0.996i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.0871 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.826564661\)
\(L(\frac12)\) \(\approx\) \(1.826564661\)
\(L(2)\) 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 \( 1 \)
5 \( 1 + (1.93 + 1.11i)T \)
good7 \( 1 + (-5.78 - 10.0i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (2.14 - 1.23i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (2.00 - 3.46i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 13.4iT - 289T^{2} \)
19 \( 1 - 34.7T + 361T^{2} \)
23 \( 1 + (-7.48 - 4.32i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (17.1 - 9.89i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-0.0722 + 0.125i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 12.6T + 1.36e3T^{2} \)
41 \( 1 + (5.90 + 3.40i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-14.4 - 25.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-54.7 + 31.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 84.1iT - 2.80e3T^{2} \)
59 \( 1 + (88.7 + 51.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-22.3 - 38.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (14.3 - 24.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 125. iT - 5.04e3T^{2} \)
73 \( 1 + 104.T + 5.32e3T^{2} \)
79 \( 1 + (10.4 + 18.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-75.3 + 43.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 104. iT - 7.92e3T^{2} \)
97 \( 1 + (-36.6 - 63.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.090573464842116950444357833553, −8.850457455919041955916257349965, −7.67451847618994727002628246705, −7.31148344453029671763273055485, −5.91230240310405935310474140222, −5.26333838826236448914819869438, −4.63113762881145401994661598783, −3.28520013263205008496701637728, −2.38601389986558549994845920933, −1.19221746969793758498920467483, 0.55281113101939648785157225270, 1.61833961617335752416209603727, 3.13170620962428207431029317609, 3.93357129572060424208799222860, 4.78649325727972349340071677201, 5.66487194825312003262323715084, 6.83247642578110208142710425514, 7.61484125219086150375566074442, 7.88632708985293806897655750997, 8.992095366185899583295791993059

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