Properties

Label 2-45-5.3-c2-0-3
Degree $2$
Conductor $45$
Sign $0.437 + 0.899i$
Analytic cond. $1.22616$
Root an. cond. $1.10732$
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  + (2.22 − 2.22i)2-s − 5.89i·4-s + (−2.67 + 4.22i)5-s + (−1.44 + 1.44i)7-s + (−4.22 − 4.22i)8-s + (3.44 + 15.3i)10-s + 3.34·11-s + (−10.4 − 10.4i)13-s + 6.44i·14-s + 4.79·16-s + (2.65 − 2.65i)17-s + 20.6i·19-s + (24.9 + 15.7i)20-s + (7.44 − 7.44i)22-s + (−16.4 − 16.4i)23-s + ⋯
L(s)  = 1  + (1.11 − 1.11i)2-s − 1.47i·4-s + (−0.534 + 0.844i)5-s + (−0.207 + 0.207i)7-s + (−0.528 − 0.528i)8-s + (0.344 + 1.53i)10-s + 0.304·11-s + (−0.803 − 0.803i)13-s + 0.460i·14-s + 0.299·16-s + (0.155 − 0.155i)17-s + 1.08i·19-s + (1.24 + 0.788i)20-s + (0.338 − 0.338i)22-s + (−0.715 − 0.715i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.437 + 0.899i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ 0.437 + 0.899i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.41486 - 0.884982i\)
\(L(\frac12)\) \(\approx\) \(1.41486 - 0.884982i\)
\(L(2)\) 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 \( 1 + (2.67 - 4.22i)T \)
good2 \( 1 + (-2.22 + 2.22i)T - 4iT^{2} \)
7 \( 1 + (1.44 - 1.44i)T - 49iT^{2} \)
11 \( 1 - 3.34T + 121T^{2} \)
13 \( 1 + (10.4 + 10.4i)T + 169iT^{2} \)
17 \( 1 + (-2.65 + 2.65i)T - 289iT^{2} \)
19 \( 1 - 20.6iT - 361T^{2} \)
23 \( 1 + (16.4 + 16.4i)T + 529iT^{2} \)
29 \( 1 - 0.853iT - 841T^{2} \)
31 \( 1 + 18.6T + 961T^{2} \)
37 \( 1 + (-38.0 + 38.0i)T - 1.36e3iT^{2} \)
41 \( 1 - 28.6T + 1.68e3T^{2} \)
43 \( 1 + (-22.4 - 22.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (19.7 - 19.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (28.6 + 28.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 111. iT - 3.48e3T^{2} \)
61 \( 1 - 94.0T + 3.72e3T^{2} \)
67 \( 1 + (54.8 - 54.8i)T - 4.48e3iT^{2} \)
71 \( 1 - 68T + 5.04e3T^{2} \)
73 \( 1 + (39.7 + 39.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (-21.1 - 21.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 94.1iT - 7.92e3T^{2} \)
97 \( 1 + (-14.5 + 14.5i)T - 9.40e3iT^{2} \)
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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

−14.78777163689253696853119514978, −14.32024037263950906768715819308, −12.79292057528765604776159924703, −12.03210438982907241744861421975, −10.92185659550895205759178113033, −9.932390179683085489899533334420, −7.72991959115636578419192481699, −5.87556272708633580394221834634, −4.09524338616557848750338902898, −2.70560374470531297517706844904, 4.00404658036720457325105214560, 5.15347458764630957845446569214, 6.74213050607634281568134061313, 7.906551238530555087140884762061, 9.454685288103260934430922298558, 11.61319135221186312571730352533, 12.67573348063870124180431544918, 13.63620005416769260220052926819, 14.73415193297376754579948161406, 15.74480818719965891638636147491

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