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Properties

Label 2-45-45.4-c3-0-13
Degree $2$
Conductor $45$
Sign $0.202 + 0.979i$
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  + (4.37 − 2.52i)2-s + (−5.16 − 0.559i)3-s + (8.77 − 15.1i)4-s + (10.8 − 2.71i)5-s + (−24.0 + 10.6i)6-s + (−18.1 + 10.5i)7-s − 48.2i·8-s + (26.3 + 5.78i)9-s + (40.6 − 39.2i)10-s + (14.2 + 24.7i)11-s + (−53.8 + 73.5i)12-s + (8.72 + 5.03i)13-s + (−53.1 + 91.9i)14-s + (−57.5 + 7.93i)15-s + (−51.7 − 89.6i)16-s + 82.7i·17-s + ⋯
L(s)  = 1  + (1.54 − 0.893i)2-s + (−0.994 − 0.107i)3-s + (1.09 − 1.89i)4-s + (0.970 − 0.242i)5-s + (−1.63 + 0.721i)6-s + (−0.982 + 0.567i)7-s − 2.13i·8-s + (0.976 + 0.214i)9-s + (1.28 − 1.24i)10-s + (0.391 + 0.678i)11-s + (−1.29 + 1.77i)12-s + (0.186 + 0.107i)13-s + (−1.01 + 1.75i)14-s + (−0.990 + 0.136i)15-s + (−0.808 − 1.40i)16-s + 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.202 + 0.979i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.202 + 0.979i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.80462 - 1.46994i\)
\(L(\frac12)\) \(\approx\) \(1.80462 - 1.46994i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.16 + 0.559i)T \)
5 \( 1 + (-10.8 + 2.71i)T \)
good2 \( 1 + (-4.37 + 2.52i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (18.1 - 10.5i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-14.2 - 24.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-8.72 - 5.03i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 82.7iT - 4.91e3T^{2} \)
19 \( 1 + 1.91T + 6.85e3T^{2} \)
23 \( 1 + (147. + 85.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (128. + 222. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (24.0 - 41.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 161. iT - 5.06e4T^{2} \)
41 \( 1 + (139. - 241. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-233. + 134. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-8.30 + 4.79i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 35.7iT - 1.48e5T^{2} \)
59 \( 1 + (281. - 487. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-39.6 - 68.6i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-404. - 233. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 316.T + 3.57e5T^{2} \)
73 \( 1 + 633. iT - 3.89e5T^{2} \)
79 \( 1 + (395. + 685. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-197. + 114. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 53.9T + 7.04e5T^{2} \)
97 \( 1 + (-83.7 + 48.3i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.81297864500951198860883585695, −13.45989992408266307912570791822, −12.66276310278491149011400222704, −12.02844840721151107079885213576, −10.58472968303159703446062770232, −9.684308305144139671534768689897, −6.35235862864338145230553594908, −5.78554569855625078870649885268, −4.21877259735353131480783391862, −1.99923764033196621490645994342, 3.57633455957613299228725377433, 5.30945574327203052722357706220, 6.25745776135149598544050941915, 7.12811311885383503693896938053, 9.713916089858742956545940312473, 11.21153210058086752339917294110, 12.51348505968389043661239782754, 13.46044840317340609437208635547, 14.18151555564438431727697878536, 15.73352822213414571452382539653

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