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Properties

Label 2-45-5.3-c8-0-4
Degree $2$
Conductor $45$
Sign $0.851 - 0.524i$
Analytic cond. $18.3320$
Root an. cond. $4.28159$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−15.2 + 15.2i)2-s − 209. i·4-s + (−558. − 280. i)5-s + (−2.41e3 + 2.41e3i)7-s + (−705. − 705. i)8-s + (1.28e4 − 4.24e3i)10-s + 981.·11-s + (−2.65e4 − 2.65e4i)13-s − 7.37e4i·14-s + 7.52e4·16-s + (1.85e4 − 1.85e4i)17-s + 5.03e4i·19-s + (−5.88e4 + 1.17e5i)20-s + (−1.49e4 + 1.49e4i)22-s + (−1.36e4 − 1.36e4i)23-s + ⋯
L(s)  = 1  + (−0.953 + 0.953i)2-s − 0.819i·4-s + (−0.893 − 0.448i)5-s + (−1.00 + 1.00i)7-s + (−0.172 − 0.172i)8-s + (1.28 − 0.424i)10-s + 0.0670·11-s + (−0.930 − 0.930i)13-s − 1.91i·14-s + 1.14·16-s + (0.221 − 0.221i)17-s + 0.386i·19-s + (−0.367 + 0.732i)20-s + (−0.0639 + 0.0639i)22-s + (−0.0487 − 0.0487i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.851 - 0.524i$
Analytic conductor: \(18.3320\)
Root analytic conductor: \(4.28159\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :4),\ 0.851 - 0.524i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.465609 + 0.131842i\)
\(L(\frac12)\) \(\approx\) \(0.465609 + 0.131842i\)
\(L(5)\) 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 \( 1 + (558. + 280. i)T \)
good2 \( 1 + (15.2 - 15.2i)T - 256iT^{2} \)
7 \( 1 + (2.41e3 - 2.41e3i)T - 5.76e6iT^{2} \)
11 \( 1 - 981.T + 2.14e8T^{2} \)
13 \( 1 + (2.65e4 + 2.65e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (-1.85e4 + 1.85e4i)T - 6.97e9iT^{2} \)
19 \( 1 - 5.03e4iT - 1.69e10T^{2} \)
23 \( 1 + (1.36e4 + 1.36e4i)T + 7.83e10iT^{2} \)
29 \( 1 - 1.05e6iT - 5.00e11T^{2} \)
31 \( 1 - 1.09e6T + 8.52e11T^{2} \)
37 \( 1 + (-7.78e4 + 7.78e4i)T - 3.51e12iT^{2} \)
41 \( 1 + 5.54e5T + 7.98e12T^{2} \)
43 \( 1 + (1.07e6 + 1.07e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (-4.06e6 + 4.06e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (1.88e6 + 1.88e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 1.27e7iT - 1.46e14T^{2} \)
61 \( 1 - 1.40e7T + 1.91e14T^{2} \)
67 \( 1 + (9.54e6 - 9.54e6i)T - 4.06e14iT^{2} \)
71 \( 1 - 2.82e7T + 6.45e14T^{2} \)
73 \( 1 + (-1.11e7 - 1.11e7i)T + 8.06e14iT^{2} \)
79 \( 1 + 6.87e7iT - 1.51e15T^{2} \)
83 \( 1 + (-3.29e6 - 3.29e6i)T + 2.25e15iT^{2} \)
89 \( 1 + 7.97e7iT - 3.93e15T^{2} \)
97 \( 1 + (1.96e7 - 1.96e7i)T - 7.83e15iT^{2} \)
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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

−14.83208807289019143308978453328, −12.72873661782508272379689923033, −12.07125589683585084007536976402, −10.08296921596843909220185798171, −9.000125586807615791738581226245, −8.039346781134232793752343893217, −6.85135918345786827736108307359, −5.40026336373054847065532106255, −3.19390325903650332216816295279, −0.43502008096455994607391502958, 0.67754169716033985306631437015, 2.68103376054177015661813175788, 4.08022893680600692059266699379, 6.68470865271232699048368026348, 7.917791441373339155196408281176, 9.479801321615129752135598876226, 10.30349740638599020861944781244, 11.41876653045541415944367511983, 12.34156031447231783923353809650, 13.88610687450611604123674342098

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