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Properties

Label 2-45-45.14-c4-0-21
Degree $2$
Conductor $45$
Sign $0.393 - 0.919i$
Analytic cond. $4.65164$
Root an. cond. $2.15676$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.12 − 5.41i)2-s + (−7.73 − 4.59i)3-s + (−11.5 + 19.9i)4-s + (−11.8 − 22.0i)5-s + (−0.705 + 56.2i)6-s + (−10.1 + 5.84i)7-s + 43.8·8-s + (38.7 + 71.1i)9-s + (−82.1 + 132. i)10-s + (141. − 81.6i)11-s + (180. − 101. i)12-s + (48.3 + 27.9i)13-s + (63.2 + 36.5i)14-s + (−9.70 + 224. i)15-s + (47.1 + 81.6i)16-s − 256.·17-s + ⋯
L(s)  = 1  + (−0.780 − 1.35i)2-s + (−0.859 − 0.510i)3-s + (−0.719 + 1.24i)4-s + (−0.473 − 0.880i)5-s + (−0.0195 + 1.56i)6-s + (−0.206 + 0.119i)7-s + 0.685·8-s + (0.478 + 0.878i)9-s + (−0.821 + 1.32i)10-s + (1.16 − 0.675i)11-s + (1.25 − 0.703i)12-s + (0.286 + 0.165i)13-s + (0.322 + 0.186i)14-s + (−0.0431 + 0.999i)15-s + (0.184 + 0.318i)16-s − 0.888·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.393 - 0.919i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.393 - 0.919i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.108999 + 0.0718822i\)
\(L(\frac12)\) \(\approx\) \(0.108999 + 0.0718822i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (7.73 + 4.59i)T \)
5 \( 1 + (11.8 + 22.0i)T \)
good2 \( 1 + (3.12 + 5.41i)T + (-8 + 13.8i)T^{2} \)
7 \( 1 + (10.1 - 5.84i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-141. + 81.6i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-48.3 - 27.9i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 256.T + 8.35e4T^{2} \)
19 \( 1 + 618.T + 1.30e5T^{2} \)
23 \( 1 + (329. - 570. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-122. + 70.8i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-300. + 519. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 751. iT - 1.87e6T^{2} \)
41 \( 1 + (2.03e3 + 1.17e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-1.71e3 + 993. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-433. - 751. i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 3.19e3T + 7.89e6T^{2} \)
59 \( 1 + (2.39e3 + 1.38e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-48.4 - 83.8i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-3.75e3 - 2.16e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 8.11e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.09e3iT - 2.83e7T^{2} \)
79 \( 1 + (929. + 1.61e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (6.08e3 + 1.05e4i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 1.06e4iT - 6.27e7T^{2} \)
97 \( 1 + (-5.62e3 + 3.24e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−13.46611226205269998044774361632, −12.44918465301445854015844453602, −11.65682595501775279881181809438, −10.85305855379094810654305189205, −9.298068060983492547401939605395, −8.272722936313073269869094804422, −6.25679976315231086323845559081, −4.11602097699517089599688882675, −1.61165812945081138787312650638, −0.12136552773282023270991851785, 4.28498635341307190323962165729, 6.37298613387372681036769059348, 6.79371574834844482072914602934, 8.525358101863779296539381387761, 9.871826903676387068836245014223, 10.96120078040451050038150418073, 12.33741209002879551918115863762, 14.47895537966735898251684262328, 15.17935356923929695791208319035, 16.05992986151669301495871947582

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