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Properties

Label 2-45-5.4-c21-0-10
Degree $2$
Conductor $45$
Sign $0.957 + 0.287i$
Analytic cond. $125.764$
Root an. cond. $11.2144$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.08e3i·2-s + 9.24e5·4-s + (−2.09e7 − 6.27e6i)5-s − 7.89e8i·7-s − 3.27e9i·8-s + (−6.79e9 + 2.26e10i)10-s − 1.26e11·11-s + 8.10e10i·13-s − 8.55e11·14-s − 1.60e12·16-s + 1.14e13i·17-s + 3.33e13·19-s + (−1.93e13 − 5.80e12i)20-s + 1.37e14i·22-s + 2.69e14i·23-s + ⋯
L(s)  = 1  − 0.747i·2-s + 0.440·4-s + (−0.957 − 0.287i)5-s − 1.05i·7-s − 1.07i·8-s + (−0.214 + 0.716i)10-s − 1.47·11-s + 0.162i·13-s − 0.790·14-s − 0.365·16-s + 1.37i·17-s + 1.24·19-s + (−0.422 − 0.126i)20-s + 1.10i·22-s + 1.35i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.957 + 0.287i$
Analytic conductor: \(125.764\)
Root analytic conductor: \(11.2144\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :21/2),\ 0.957 + 0.287i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.236477941\)
\(L(\frac12)\) \(\approx\) \(1.236477941\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2.09e7 + 6.27e6i)T \)
good2 \( 1 + 1.08e3iT - 2.09e6T^{2} \)
7 \( 1 + 7.89e8iT - 5.58e17T^{2} \)
11 \( 1 + 1.26e11T + 7.40e21T^{2} \)
13 \( 1 - 8.10e10iT - 2.47e23T^{2} \)
17 \( 1 - 1.14e13iT - 6.90e25T^{2} \)
19 \( 1 - 3.33e13T + 7.14e26T^{2} \)
23 \( 1 - 2.69e14iT - 3.94e28T^{2} \)
29 \( 1 + 4.24e15T + 5.13e30T^{2} \)
31 \( 1 + 4.38e15T + 2.08e31T^{2} \)
37 \( 1 + 2.00e16iT - 8.55e32T^{2} \)
41 \( 1 - 1.19e17T + 7.38e33T^{2} \)
43 \( 1 - 1.96e17iT - 2.00e34T^{2} \)
47 \( 1 + 3.56e17iT - 1.30e35T^{2} \)
53 \( 1 + 1.03e18iT - 1.62e36T^{2} \)
59 \( 1 + 4.27e17T + 1.54e37T^{2} \)
61 \( 1 + 5.02e18T + 3.10e37T^{2} \)
67 \( 1 - 1.24e19iT - 2.22e38T^{2} \)
71 \( 1 - 3.25e19T + 7.52e38T^{2} \)
73 \( 1 + 2.51e19iT - 1.34e39T^{2} \)
79 \( 1 + 1.33e19T + 7.08e39T^{2} \)
83 \( 1 - 1.80e20iT - 1.99e40T^{2} \)
89 \( 1 - 1.47e20T + 8.65e40T^{2} \)
97 \( 1 - 5.24e20iT - 5.27e41T^{2} \)
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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

−11.29382044131883989605410089535, −10.77936808961404015010097045861, −9.586050961462227125015590428358, −7.76577572797303917775247845032, −7.32757758181241276266874559563, −5.53116869711687296814497138145, −3.97762098190260681846724360341, −3.31847917172619596678776634173, −1.82055141869147351042842425606, −0.75091016741644136640641925499, 0.32830806412763142002435381104, 2.33360172599746250005720455396, 3.05883581157556036082030395457, 4.96010293258282553343215801766, 5.78114146313625461801324126901, 7.27785463988724284890396056220, 7.80839882191139884004271252743, 9.063541511385006722697745725098, 10.75814618542531944922469121661, 11.62300952329858462917831396943

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