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Properties

Label 2-45-45.4-c1-0-0
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  + (−1.67 + 0.965i)2-s + (1.67 − 0.448i)3-s + (0.866 − 1.50i)4-s + (0.358 + 2.20i)5-s + (−2.36 + 2.36i)6-s + (−0.776 + 0.448i)7-s − 0.517i·8-s + (2.59 − 1.50i)9-s + (−2.73 − 3.34i)10-s + (−2.36 − 4.09i)11-s + (0.776 − 2.89i)12-s + (−2.12 − 1.22i)13-s + (0.866 − 1.50i)14-s + (1.58 + 3.53i)15-s + (2.23 + 3.86i)16-s − 0.378i·17-s + ⋯
L(s)  = 1  + (−1.18 + 0.683i)2-s + (0.965 − 0.258i)3-s + (0.433 − 0.750i)4-s + (0.160 + 0.987i)5-s + (−0.965 + 0.965i)6-s + (−0.293 + 0.169i)7-s − 0.183i·8-s + (0.866 − 0.5i)9-s + (−0.863 − 1.05i)10-s + (−0.713 − 1.23i)11-s + (0.224 − 0.836i)12-s + (−0.588 − 0.339i)13-s + (0.231 − 0.400i)14-s + (0.410 + 0.911i)15-s + (0.558 + 0.966i)16-s − 0.0919i·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

\(L(1)\) \(\approx\) \(0.544768 + 0.249595i\)
\(L(\frac12)\) \(\approx\) \(0.544768 + 0.249595i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 + 0.448i)T \)
5 \( 1 + (-0.358 - 2.20i)T \)
good2 \( 1 + (1.67 - 0.965i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (0.776 - 0.448i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.36 + 4.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.12 + 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.378iT - 17T^{2} \)
19 \( 1 + 2.73T + 19T^{2} \)
23 \( 1 + (-1.67 - 0.965i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.23 - 5.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.36 + 2.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.24iT - 37T^{2} \)
41 \( 1 + (2.13 - 3.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.91 + 4.57i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.79 - 2.19i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.86iT - 53T^{2} \)
59 \( 1 + (-1.26 + 2.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.33 + 9.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.45 + 2.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.80T + 71T^{2} \)
73 \( 1 - 8.48iT - 73T^{2} \)
79 \( 1 + (-0.267 - 0.464i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.02 - 5.20i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.39T + 89T^{2} \)
97 \( 1 + (-8.90 + 5.13i)T + (48.5 - 84.0i)T^{2} \)
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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

−15.95227049926260134960028610510, −15.15702209744160000080736239296, −14.01295886335869545953171481701, −12.81587397374219572394116198279, −10.74774857570415806053232201862, −9.700221835786415295252952629243, −8.502430978922669459159088065219, −7.51889093204330101270140991174, −6.34447043357801617997453194865, −3.05579008791127064836533074837, 2.20090208401029123785420405987, 4.67693449825763920894240738022, 7.55391847213597745140673027120, 8.671557550900203574482534551111, 9.635977353063991450683091271177, 10.36056186085278500567815859486, 12.19130910616500093383123252507, 13.21825836752224566816989682590, 14.65610490126536639919263559664, 15.92421242436458489356493225576

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