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Properties

Label 2-45-45.2-c1-0-0
Degree $2$
Conductor $45$
Sign $0.335 - 0.942i$
Analytic cond. $0.359326$
Root an. cond. $0.599438$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.347 + 1.29i)2-s + (−1.18 + 1.25i)3-s + (0.170 − 0.0981i)4-s + (−1.59 − 1.56i)5-s + (−2.04 − 1.10i)6-s + (1.97 − 0.530i)7-s + (2.08 + 2.08i)8-s + (−0.170 − 2.99i)9-s + (1.48 − 2.61i)10-s + (−0.762 − 0.440i)11-s + (−0.0786 + 0.330i)12-s + (−5.36 − 1.43i)13-s + (1.37 + 2.38i)14-s + (3.87 − 0.140i)15-s + (−1.78 + 3.09i)16-s + (−1.13 + 1.13i)17-s + ⋯
L(s)  = 1  + (0.245 + 0.917i)2-s + (−0.686 + 0.726i)3-s + (0.0850 − 0.0490i)4-s + (−0.712 − 0.701i)5-s + (−0.835 − 0.451i)6-s + (0.747 − 0.200i)7-s + (0.737 + 0.737i)8-s + (−0.0566 − 0.998i)9-s + (0.468 − 0.826i)10-s + (−0.229 − 0.132i)11-s + (−0.0227 + 0.0955i)12-s + (−1.48 − 0.398i)13-s + (0.367 + 0.636i)14-s + (0.999 − 0.0363i)15-s + (−0.446 + 0.772i)16-s + (−0.275 + 0.275i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.335 - 0.942i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ 0.335 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.637249 + 0.449727i\)
\(L(\frac12)\) \(\approx\) \(0.637249 + 0.449727i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.18 - 1.25i)T \)
5 \( 1 + (1.59 + 1.56i)T \)
good2 \( 1 + (-0.347 - 1.29i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (-1.97 + 0.530i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.762 + 0.440i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.36 + 1.43i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.13 - 1.13i)T - 17iT^{2} \)
19 \( 1 - 1.52iT - 19T^{2} \)
23 \( 1 + (-0.410 + 1.53i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.796 + 1.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.49 - 6.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.25 + 4.25i)T + 37iT^{2} \)
41 \( 1 + (-3.11 + 1.79i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.497 - 1.85i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.14 + 7.99i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.65 - 4.65i)T + 53iT^{2} \)
59 \( 1 + (-3.81 - 6.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.64 + 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.859 - 3.20i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.89iT - 71T^{2} \)
73 \( 1 + (-1.58 + 1.58i)T - 73iT^{2} \)
79 \( 1 + (6.69 + 3.86i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.59 - 2.57i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 4.62T + 89T^{2} \)
97 \( 1 + (-3.82 + 1.02i)T + (84.0 - 48.5i)T^{2} \)
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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

−16.02469853190150777759259864620, −15.20691192613173736245995783314, −14.34886667966410842395486522924, −12.47597809362197005981148012920, −11.39971997199377264959645947220, −10.26066924461989647425472729830, −8.437537083594077048053543962984, −7.17681889691893007898273825842, −5.42372865323891128466115882957, −4.54158727975833086853461579263, 2.43624834061761046528913729151, 4.70074455770456811411984524595, 6.89667870656678625652436940339, 7.79069444815478277579713790060, 10.19994729235784242419821388403, 11.43273068946636620411966783079, 11.78523451982725194939072850195, 12.93058776393251133046091413971, 14.31567133925220262811901290975, 15.64310033010143343987541305332

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