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Properties

Label 2-45-45.2-c1-0-2
Degree $2$
Conductor $45$
Sign $0.621 + 0.783i$
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  + (−0.430 − 1.60i)2-s + (1.35 + 1.08i)3-s + (−0.661 + 0.382i)4-s + (−2.23 + 0.154i)5-s + (1.15 − 2.63i)6-s + (−1.73 + 0.465i)7-s + (−1.45 − 1.45i)8-s + (0.661 + 2.92i)9-s + (1.20 + 3.51i)10-s + (3.12 + 1.80i)11-s + (−1.30 − 0.198i)12-s + (−1.27 − 0.342i)13-s + (1.49 + 2.59i)14-s + (−3.18 − 2.20i)15-s + (−2.47 + 4.28i)16-s + (0.277 − 0.277i)17-s + ⋯
L(s)  = 1  + (−0.304 − 1.13i)2-s + (0.781 + 0.624i)3-s + (−0.330 + 0.191i)4-s + (−0.997 + 0.0690i)5-s + (0.471 − 1.07i)6-s + (−0.656 + 0.175i)7-s + (−0.513 − 0.513i)8-s + (0.220 + 0.975i)9-s + (0.381 + 1.11i)10-s + (0.942 + 0.544i)11-s + (−0.377 − 0.0573i)12-s + (−0.354 − 0.0950i)13-s + (0.399 + 0.692i)14-s + (−0.822 − 0.568i)15-s + (−0.618 + 1.07i)16-s + (0.0671 − 0.0671i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.621 + 0.783i$
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: \((2,\ 45,\ (\ :1/2),\ 0.621 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692211 - 0.334449i\)
\(L(\frac12)\) \(\approx\) \(0.692211 - 0.334449i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.35 - 1.08i)T \)
5 \( 1 + (2.23 - 0.154i)T \)
good2 \( 1 + (0.430 + 1.60i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (1.73 - 0.465i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.12 - 1.80i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.27 + 0.342i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.277 + 0.277i)T - 17iT^{2} \)
19 \( 1 + 6.25iT - 19T^{2} \)
23 \( 1 + (-0.579 + 2.16i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.56 - 2.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.42 + 4.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.55 - 5.55i)T + 37iT^{2} \)
41 \( 1 + (-1.29 + 0.744i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.10 + 4.10i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.02 + 3.82i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-7.48 - 7.48i)T + 53iT^{2} \)
59 \( 1 + (-0.279 - 0.483i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.96 - 5.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.90 - 10.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.01iT - 71T^{2} \)
73 \( 1 + (1.29 - 1.29i)T - 73iT^{2} \)
79 \( 1 + (-6.96 - 4.02i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.560 - 0.150i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + (-5.14 + 1.37i)T + (84.0 - 48.5i)T^{2} \)
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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

−15.48814334931586089528425035668, −14.84401014764592595659853185392, −13.11617539726927643038549088056, −11.97751315480468378099664596534, −10.92934913075462471574604889698, −9.699828025443930919117313866918, −8.839942659986665334564159960456, −7.05478013129401502391818272930, −4.21548998350390319370472016168, −2.86093392629714111854334407398, 3.55540741718480021542886524616, 6.25278543188853638197576661280, 7.35680509729583724471517328201, 8.267373598673132246039176657071, 9.398619374399456357120601865111, 11.61150115226788427953733667974, 12.65047593048595670642384387924, 14.21957815217478921101767747719, 14.92090090360207578431044602671, 16.09261190134310147601321615436

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