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Properties

Label 2-45-9.4-c3-0-0
Degree $2$
Conductor $45$
Sign $0.539 - 0.841i$
Analytic cond. $2.65508$
Root an. cond. $1.62944$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.87 − 3.24i)2-s + (−4.05 + 3.24i)3-s + (−3.02 + 5.24i)4-s + (−2.5 + 4.33i)5-s + (18.1 + 7.08i)6-s + (15.6 + 27.1i)7-s − 7.30·8-s + (5.92 − 26.3i)9-s + 18.7·10-s + (10.4 + 18.0i)11-s + (−4.73 − 31.0i)12-s + (−29.9 + 51.9i)13-s + (58.7 − 101. i)14-s + (−3.91 − 25.6i)15-s + (37.8 + 65.6i)16-s − 74.0·17-s + ⋯
L(s)  = 1  + (−0.662 − 1.14i)2-s + (−0.780 + 0.624i)3-s + (−0.378 + 0.655i)4-s + (−0.223 + 0.387i)5-s + (1.23 + 0.482i)6-s + (0.846 + 1.46i)7-s − 0.322·8-s + (0.219 − 0.975i)9-s + 0.592·10-s + (0.285 + 0.494i)11-s + (−0.113 − 0.747i)12-s + (−0.639 + 1.10i)13-s + (1.12 − 1.94i)14-s + (−0.0673 − 0.442i)15-s + (0.592 + 1.02i)16-s − 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.539 - 0.841i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.539 - 0.841i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.489082 + 0.267372i\)
\(L(\frac12)\) \(\approx\) \(0.489082 + 0.267372i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.05 - 3.24i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
good2 \( 1 + (1.87 + 3.24i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-15.6 - 27.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-10.4 - 18.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (29.9 - 51.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 74.0T + 4.91e3T^{2} \)
19 \( 1 + 63.8T + 6.85e3T^{2} \)
23 \( 1 + (-16.4 + 28.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-80.0 - 138. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-127. + 220. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 215.T + 5.06e4T^{2} \)
41 \( 1 + (70.8 - 122. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (68.9 + 119. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (16.7 + 29.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 41.9T + 1.48e5T^{2} \)
59 \( 1 + (307. - 532. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-67.1 - 116. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-428. + 742. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 588.T + 3.57e5T^{2} \)
73 \( 1 + 618.T + 3.89e5T^{2} \)
79 \( 1 + (-172. - 299. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-546. - 946. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 414.T + 7.04e5T^{2} \)
97 \( 1 + (-100. - 174. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.35764160361634160069584554716, −14.79585893286629569408204543185, −12.38509043940459619458740043904, −11.67233495560359753251896719279, −10.97623768218919830589676682747, −9.630931514163198316417603448057, −8.723883964749828005219153453434, −6.36374215188516684101746101515, −4.56382228414181258301496955753, −2.24094512242362626547816312287, 0.59897692555826485278736161977, 4.81833665863422240967538581525, 6.44744065953783629708885769348, 7.55263838824873242116065683646, 8.359100912357928686243518909743, 10.33686375952297343963509173402, 11.50519502529482874697286618612, 12.93463114672094613854605231221, 14.15009216882154721277720229323, 15.51689102109445294206836207967

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