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Properties

Label 2-45-45.4-c3-0-2
Degree $2$
Conductor $45$
Sign $-0.526 - 0.849i$
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  + (0.410 − 0.236i)2-s + (−4.58 − 2.45i)3-s + (−3.88 + 6.73i)4-s + (−2.29 + 10.9i)5-s + (−2.46 + 0.0785i)6-s + (−7.10 + 4.10i)7-s + 7.47i·8-s + (14.9 + 22.4i)9-s + (1.64 + 5.03i)10-s + (−2.63 − 4.56i)11-s + (34.3 − 21.3i)12-s + (−66.7 − 38.5i)13-s + (−1.94 + 3.36i)14-s + (37.3 − 44.4i)15-s + (−29.3 − 50.7i)16-s + 88.9i·17-s + ⋯
L(s)  = 1  + (0.145 − 0.0837i)2-s + (−0.881 − 0.472i)3-s + (−0.485 + 0.841i)4-s + (−0.205 + 0.978i)5-s + (−0.167 + 0.00534i)6-s + (−0.383 + 0.221i)7-s + 0.330i·8-s + (0.554 + 0.832i)9-s + (0.0521 + 0.159i)10-s + (−0.0721 − 0.124i)11-s + (0.825 − 0.512i)12-s + (−1.42 − 0.821i)13-s + (−0.0371 + 0.0643i)14-s + (0.643 − 0.765i)15-s + (−0.458 − 0.793i)16-s + 1.26i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.849i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.526 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.526 - 0.849i$
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: \((2,\ 45,\ (\ :3/2),\ -0.526 - 0.849i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.294833 + 0.529653i\)
\(L(\frac12)\) \(\approx\) \(0.294833 + 0.529653i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.58 + 2.45i)T \)
5 \( 1 + (2.29 - 10.9i)T \)
good2 \( 1 + (-0.410 + 0.236i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (7.10 - 4.10i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (2.63 + 4.56i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (66.7 + 38.5i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 88.9iT - 4.91e3T^{2} \)
19 \( 1 - 91.7T + 6.85e3T^{2} \)
23 \( 1 + (-133. - 77.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-84.1 - 145. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (36.3 - 63.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 154. iT - 5.06e4T^{2} \)
41 \( 1 + (3.97 - 6.88i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (20.0 - 11.6i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (261. - 150. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 344. iT - 1.48e5T^{2} \)
59 \( 1 + (-125. + 217. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (136. + 236. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-723. - 417. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 351.T + 3.57e5T^{2} \)
73 \( 1 - 522. iT - 3.89e5T^{2} \)
79 \( 1 + (350. + 606. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-208. + 120. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + (-337. + 194. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−15.83001497315218997697018438197, −14.49166428332148605403681664332, −13.08769812628163137732263926210, −12.31904138203499198021341545911, −11.21278095443297602315117463201, −9.912086885984607822585992274595, −7.926970728366036042466138497763, −6.90353617756996939256396219060, −5.18948849363452719745807723304, −3.11197117343296550430472371583, 0.51551481074986075853014129011, 4.51901627015293375716642089244, 5.29993328969757560486552868240, 6.99091989872853978256540285661, 9.300485302954038483189677835089, 9.855652593957709434833809626527, 11.46840387465516170941451996234, 12.52168474915448679890377112158, 13.79969575095510535448261164691, 15.11668639468082645407571559367

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