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Properties

Label 2-45-45.38-c1-0-3
Degree $2$
Conductor $45$
Sign $0.914 + 0.405i$
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  + (1.29 − 0.347i)2-s + (−1.25 − 1.18i)3-s + (−0.170 + 0.0981i)4-s + (0.561 + 2.16i)5-s + (−2.04 − 1.10i)6-s + (−0.530 − 1.97i)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 + (0.330 + 0.0786i)12-s + (1.43 − 5.36i)13-s + (−1.37 − 2.38i)14-s + (1.86 − 3.39i)15-s + (−1.78 + 3.09i)16-s + (1.13 + 1.13i)17-s + ⋯
L(s)  = 1  + (0.917 − 0.245i)2-s + (−0.726 − 0.686i)3-s + (−0.0850 + 0.0490i)4-s + (0.251 + 0.967i)5-s + (−0.835 − 0.451i)6-s + (−0.200 − 0.747i)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 + (0.0955 + 0.0227i)12-s + (0.398 − 1.48i)13-s + (−0.367 − 0.636i)14-s + (0.482 − 0.876i)15-s + (−0.446 + 0.772i)16-s + (0.275 + 0.275i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.914 + 0.405i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ 0.914 + 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908795 - 0.192553i\)
\(L(\frac12)\) \(\approx\) \(0.908795 - 0.192553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.25 + 1.18i)T \)
5 \( 1 + (-0.561 - 2.16i)T \)
good2 \( 1 + (-1.29 + 0.347i)T + (1.73 - i)T^{2} \)
7 \( 1 + (0.530 + 1.97i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.762 + 0.440i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.43 + 5.36i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.13 - 1.13i)T + 17iT^{2} \)
19 \( 1 + 1.52iT - 19T^{2} \)
23 \( 1 + (-1.53 - 0.410i)T + (19.9 + 11.5i)T^{2} \)
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 \( 1 + (1.85 - 0.497i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (7.99 - 2.14i)T + (40.7 - 23.5i)T^{2} \)
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 \( 1 + (-3.20 - 0.859i)T + (58.0 + 33.5i)T^{2} \)
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 \( 1 + (2.57 + 9.59i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 4.62T + 89T^{2} \)
97 \( 1 + (1.02 + 3.82i)T + (-84.0 + 48.5i)T^{2} \)
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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

−15.59657646459096035514199974599, −14.24236521316121230296315190850, −13.38199765095037875797972014176, −12.59499484701945719157699524994, −11.21384684316175528155496193052, −10.33562473523245126863328887327, −7.995009914421974129313984041244, −6.54960885604503969738464840649, −5.26479471397507174895390008046, −3.20535998862249675067561604912, 4.16316182600206935947866334557, 5.28376003263830924880713587516, 6.30911524792698643217080003303, 8.958020655315938826328372400270, 9.749414230885313884008189789245, 11.65526650813513380772443743975, 12.48495306667605603154938200310, 13.64061678458127921857573149882, 14.89762709726556920221229562151, 15.93450609007233357183940895798

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