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Properties

Label 2-45-45.32-c1-0-0
Degree $2$
Conductor $45$
Sign $0.835 - 0.549i$
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  + (−2.24 − 0.601i)2-s + (0.173 + 1.72i)3-s + (2.93 + 1.69i)4-s + (2.10 + 0.759i)5-s + (0.647 − 3.97i)6-s + (−0.201 + 0.751i)7-s + (−2.29 − 2.29i)8-s + (−2.93 + 0.597i)9-s + (−4.26 − 2.96i)10-s + (−0.220 + 0.127i)11-s + (−2.41 + 5.36i)12-s + (−0.992 − 3.70i)13-s + (0.903 − 1.56i)14-s + (−0.943 + 3.75i)15-s + (0.367 + 0.636i)16-s + (3.93 − 3.93i)17-s + ⋯
L(s)  = 1  + (−1.58 − 0.425i)2-s + (0.100 + 0.994i)3-s + (1.46 + 0.848i)4-s + (0.940 + 0.339i)5-s + (0.264 − 1.62i)6-s + (−0.0761 + 0.284i)7-s + (−0.809 − 0.809i)8-s + (−0.979 + 0.199i)9-s + (−1.34 − 0.938i)10-s + (−0.0663 + 0.0383i)11-s + (−0.697 + 1.54i)12-s + (−0.275 − 1.02i)13-s + (0.241 − 0.418i)14-s + (−0.243 + 0.969i)15-s + (0.0918 + 0.159i)16-s + (0.953 − 0.953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.835 - 0.549i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ 0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.447152 + 0.133944i\)
\(L(\frac12)\) \(\approx\) \(0.447152 + 0.133944i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.173 - 1.72i)T \)
5 \( 1 + (-2.10 - 0.759i)T \)
good2 \( 1 + (2.24 + 0.601i)T + (1.73 + i)T^{2} \)
7 \( 1 + (0.201 - 0.751i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.220 - 0.127i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.992 + 3.70i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-3.93 + 3.93i)T - 17iT^{2} \)
19 \( 1 - 0.440iT - 19T^{2} \)
23 \( 1 + (3.42 - 0.917i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.76 + 4.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0971 - 0.168i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.123 + 0.123i)T + 37iT^{2} \)
41 \( 1 + (3.88 + 2.24i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.33 - 0.357i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.17 - 1.11i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.938 - 0.938i)T + 53iT^{2} \)
59 \( 1 + (4.02 - 6.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.44 + 2.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.9 - 3.47i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.15iT - 71T^{2} \)
73 \( 1 + (9.18 - 9.18i)T - 73iT^{2} \)
79 \( 1 + (-11.9 + 6.91i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.39 - 5.20i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 0.285T + 89T^{2} \)
97 \( 1 + (2.34 - 8.73i)T + (-84.0 - 48.5i)T^{2} \)
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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

−16.35341902362430608721378046131, −15.18849341322917771569540753192, −13.86123692548464209638655087766, −11.93288751189051156972942566043, −10.60594299502576555141575910100, −9.927627682994797509922566598811, −9.102140476224895414523037219859, −7.70456680839551476316022732279, −5.60776231226021653543226294833, −2.75439607419863582690830338006, 1.70261660391969818263383751773, 6.01464448945800986668026106474, 7.14521249088469671462405737597, 8.390319490642632697487370575033, 9.413356830126716524462238479353, 10.59757369480016211620654601410, 12.20017190052507115419584291589, 13.53520364070056366168062711939, 14.67155705791519634025439956122, 16.52040301663621778150302810730

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