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Properties

Label 2-45-45.43-c2-0-0
Degree $2$
Conductor $45$
Sign $-0.751 - 0.660i$
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  + (0.863 + 3.22i)2-s + (−2.79 + 1.08i)3-s + (−6.18 + 3.57i)4-s + (3.34 − 3.71i)5-s + (−5.91 − 8.07i)6-s + (2.35 + 8.79i)7-s + (−7.41 − 7.41i)8-s + (6.64 − 6.07i)9-s + (14.8 + 7.58i)10-s + (1.02 − 1.77i)11-s + (13.4 − 16.7i)12-s + (−1.12 + 4.18i)13-s + (−26.3 + 15.2i)14-s + (−5.32 + 14.0i)15-s + (3.22 − 5.57i)16-s + (17.1 − 17.1i)17-s + ⋯
L(s)  = 1  + (0.431 + 1.61i)2-s + (−0.932 + 0.362i)3-s + (−1.54 + 0.892i)4-s + (0.669 − 0.742i)5-s + (−0.986 − 1.34i)6-s + (0.336 + 1.25i)7-s + (−0.927 − 0.927i)8-s + (0.737 − 0.675i)9-s + (1.48 + 0.758i)10-s + (0.0930 − 0.161i)11-s + (1.11 − 1.39i)12-s + (−0.0862 + 0.321i)13-s + (−1.88 + 1.08i)14-s + (−0.355 + 0.934i)15-s + (0.201 − 0.348i)16-s + (1.00 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.751 - 0.660i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ -0.751 - 0.660i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.388396 + 1.03048i\)
\(L(\frac12)\) \(\approx\) \(0.388396 + 1.03048i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.79 - 1.08i)T \)
5 \( 1 + (-3.34 + 3.71i)T \)
good2 \( 1 + (-0.863 - 3.22i)T + (-3.46 + 2i)T^{2} \)
7 \( 1 + (-2.35 - 8.79i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-1.02 + 1.77i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (1.12 - 4.18i)T + (-146. - 84.5i)T^{2} \)
17 \( 1 + (-17.1 + 17.1i)T - 289iT^{2} \)
19 \( 1 + 18.6iT - 361T^{2} \)
23 \( 1 + (-3.08 + 11.5i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + (19.4 + 11.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-11.9 - 20.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (33.2 - 33.2i)T - 1.36e3iT^{2} \)
41 \( 1 + (-2.15 - 3.73i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (30.4 - 8.15i)T + (1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-0.224 - 0.838i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (3.33 + 3.33i)T + 2.80e3iT^{2} \)
59 \( 1 + (-66.3 + 38.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-35.3 + 61.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (20.8 + 5.59i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 28.6T + 5.04e3T^{2} \)
73 \( 1 + (-48.1 - 48.1i)T + 5.32e3iT^{2} \)
79 \( 1 + (-0.201 - 0.116i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (153. - 41.1i)T + (5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 103. iT - 7.92e3T^{2} \)
97 \( 1 + (18.6 + 69.7i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−16.06449842932984685553295672830, −15.22721019158286354769578295960, −14.04481728121741665061933635814, −12.76128385706794764244384859353, −11.67064691676244259666105043419, −9.592311565297744655498154785476, −8.531449169272703395397541498730, −6.72353954905549809551611095434, −5.50993970851222542601337867606, −4.89306157923792072286120971707, 1.54069494954076268735665417130, 3.83224895122027192421891478579, 5.59826619625073914226863012582, 7.36352802416634824068050266607, 10.09330251427429626233302996020, 10.46949749220885907697633798699, 11.51009533673456263216920300540, 12.69316048972699335523474302975, 13.59449981647351281643793319639, 14.57439172888761688754609064115

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