Properties

Label 2-45-45.13-c2-0-5
Degree $2$
Conductor $45$
Sign $0.851 - 0.524i$
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  + (3.47 + 0.930i)2-s + (−2.93 + 0.635i)3-s + (7.72 + 4.46i)4-s + (−2.41 − 4.37i)5-s + (−10.7 − 0.522i)6-s + (−4.78 − 1.28i)7-s + (12.5 + 12.5i)8-s + (8.19 − 3.72i)9-s + (−4.32 − 17.4i)10-s + (1.37 + 2.38i)11-s + (−25.4 − 8.17i)12-s + (−9.15 + 2.45i)13-s + (−15.4 − 8.91i)14-s + (9.87 + 11.2i)15-s + (13.9 + 24.2i)16-s + (9.17 − 9.17i)17-s + ⋯
L(s)  = 1  + (1.73 + 0.465i)2-s + (−0.977 + 0.211i)3-s + (1.93 + 1.11i)4-s + (−0.483 − 0.875i)5-s + (−1.79 − 0.0871i)6-s + (−0.684 − 0.183i)7-s + (1.56 + 1.56i)8-s + (0.910 − 0.413i)9-s + (−0.432 − 1.74i)10-s + (0.125 + 0.216i)11-s + (−2.12 − 0.681i)12-s + (−0.704 + 0.188i)13-s + (−1.10 − 0.636i)14-s + (0.658 + 0.752i)15-s + (0.873 + 1.51i)16-s + (0.539 − 0.539i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.851 - 0.524i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ 0.851 - 0.524i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.78061 + 0.504790i\)
\(L(\frac12)\) \(\approx\) \(1.78061 + 0.504790i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.93 - 0.635i)T \)
5 \( 1 + (2.41 + 4.37i)T \)
good2 \( 1 + (-3.47 - 0.930i)T + (3.46 + 2i)T^{2} \)
7 \( 1 + (4.78 + 1.28i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-1.37 - 2.38i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (9.15 - 2.45i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (-9.17 + 9.17i)T - 289iT^{2} \)
19 \( 1 - 32.1iT - 361T^{2} \)
23 \( 1 + (0.669 - 0.179i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + (-30.1 + 17.3i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-12.8 + 22.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (13.0 - 13.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (-20.2 + 35.0i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (0.987 - 3.68i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (3.70 + 0.993i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (31.2 + 31.2i)T + 2.80e3iT^{2} \)
59 \( 1 + (-36.8 - 21.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (6.56 + 11.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-12.0 - 44.9i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 114.T + 5.04e3T^{2} \)
73 \( 1 + (18.7 + 18.7i)T + 5.32e3iT^{2} \)
79 \( 1 + (-18.3 + 10.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-3.95 + 14.7i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 92.4iT - 7.92e3T^{2} \)
97 \( 1 + (-151. - 40.7i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−15.81868782380855477002879113044, −14.58228673652223310657003201295, −13.22052535350957435845062584439, −12.24469344588415928733641181695, −11.81120604184045750204704877631, −9.961988552124005658866512124149, −7.55574368980988706492289128982, −6.20635391977831080454060593992, −5.02136552671230114467345097224, −3.88185690617760663562789592818, 3.00094119241672327225036497227, 4.70836794080559892135826508318, 6.16908153535737838021811622231, 7.07777041484096641165904550715, 10.26010726714930462568891619736, 11.20817386594474699070700437865, 12.13936734034264915111429790837, 12.96590946006348818681282359161, 14.20363814273377007054201633364, 15.34529597859921381149541627474

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