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Properties

Label 2-45-45.4-c1-0-1
Degree $2$
Conductor $45$
Sign $0.653 - 0.757i$
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  + (−0.448 + 0.258i)2-s + (0.448 + 1.67i)3-s + (−0.866 + 1.5i)4-s + (−0.358 − 2.20i)5-s + (−0.633 − 0.633i)6-s + (2.89 − 1.67i)7-s − 1.93i·8-s + (−2.59 + 1.50i)9-s + (0.732 + 0.896i)10-s + (−0.633 − 1.09i)11-s + (−2.89 − 0.776i)12-s + (−2.12 − 1.22i)13-s + (−0.866 + 1.5i)14-s + (3.53 − 1.58i)15-s + (−1.23 − 2.13i)16-s + 5.27i·17-s + ⋯
L(s)  = 1  + (−0.316 + 0.183i)2-s + (0.258 + 0.965i)3-s + (−0.433 + 0.750i)4-s + (−0.160 − 0.987i)5-s + (−0.258 − 0.258i)6-s + (1.09 − 0.632i)7-s − 0.683i·8-s + (−0.866 + 0.5i)9-s + (0.231 + 0.283i)10-s + (−0.191 − 0.331i)11-s + (−0.836 − 0.224i)12-s + (−0.588 − 0.339i)13-s + (−0.231 + 0.400i)14-s + (0.911 − 0.410i)15-s + (−0.308 − 0.533i)16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.653 - 0.757i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ 0.653 - 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640006 + 0.293230i\)
\(L(\frac12)\) \(\approx\) \(0.640006 + 0.293230i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.448 - 1.67i)T \)
5 \( 1 + (0.358 + 2.20i)T \)
good2 \( 1 + (0.448 - 0.258i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-2.89 + 1.67i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.633 + 1.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.12 + 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.27iT - 17T^{2} \)
19 \( 1 - 0.732T + 19T^{2} \)
23 \( 1 + (-0.448 - 0.258i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.232 + 0.401i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.366 - 0.633i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.24iT - 37T^{2} \)
41 \( 1 + (3.86 - 6.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.568 + 0.328i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.56 - 1.48i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.03iT - 53T^{2} \)
59 \( 1 + (-4.73 + 8.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.33 - 5.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.57 - 3.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + 8.48iT - 73T^{2} \)
79 \( 1 + (-3.73 - 6.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.90 + 3.98i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (13.1 - 7.58i)T + (48.5 - 84.0i)T^{2} \)
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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

−16.28917846486032054758719585802, −15.04520594880966610974105421384, −13.81243423877750383396943925787, −12.61318321986611336932119036402, −11.19086060461297595318524635221, −9.799972013775995612597336561948, −8.504047622228970716637441742687, −7.88515502509424774423259794821, −5.05925354367494962075770773831, −3.93576124535336025159435104885, 2.26244947342011222834222574810, 5.26400770260162682424791785396, 6.95480278415243523552271718165, 8.242180045575752546111058148941, 9.601069534051500250745263584444, 11.13133883190253041768959290440, 11.96952086924739140056256899157, 13.75302340938096798534128091001, 14.47184066230641958536629819428, 15.25765265406797709493756476543

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