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Properties

Label 2-45-5.2-c2-0-2
Degree $2$
Conductor $45$
Sign $0.793 + 0.608i$
Analytic cond. $1.22616$
Root an. cond. $1.10732$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.224 − 0.224i)2-s − 3.89i·4-s + (4.67 − 1.77i)5-s + (3.44 + 3.44i)7-s + (−1.77 + 1.77i)8-s + (−1.44 − 0.651i)10-s − 11.3·11-s + (−5.55 + 5.55i)13-s − 1.55i·14-s − 14.7·16-s + (17.3 + 17.3i)17-s + 8.69i·19-s + (−6.92 − 18.2i)20-s + (2.55 + 2.55i)22-s + (−11.5 + 11.5i)23-s + ⋯
L(s)  = 1  + (−0.112 − 0.112i)2-s − 0.974i·4-s + (0.934 − 0.355i)5-s + (0.492 + 0.492i)7-s + (−0.221 + 0.221i)8-s + (−0.144 − 0.0651i)10-s − 1.03·11-s + (−0.426 + 0.426i)13-s − 0.110i·14-s − 0.924·16-s + (1.02 + 1.02i)17-s + 0.457i·19-s + (−0.346 − 0.911i)20-s + (0.115 + 0.115i)22-s + (−0.502 + 0.502i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ 0.793 + 0.608i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.08584 - 0.368444i\)
\(L(\frac12)\) \(\approx\) \(1.08584 - 0.368444i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-4.67 + 1.77i)T \)
good2 \( 1 + (0.224 + 0.224i)T + 4iT^{2} \)
7 \( 1 + (-3.44 - 3.44i)T + 49iT^{2} \)
11 \( 1 + 11.3T + 121T^{2} \)
13 \( 1 + (5.55 - 5.55i)T - 169iT^{2} \)
17 \( 1 + (-17.3 - 17.3i)T + 289iT^{2} \)
19 \( 1 - 8.69iT - 361T^{2} \)
23 \( 1 + (11.5 - 11.5i)T - 529iT^{2} \)
29 \( 1 + 35.1iT - 841T^{2} \)
31 \( 1 - 10.6T + 961T^{2} \)
37 \( 1 + (6.04 + 6.04i)T + 1.36e3iT^{2} \)
41 \( 1 + 0.696T + 1.68e3T^{2} \)
43 \( 1 + (26.4 - 26.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (44.2 + 44.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (-0.696 + 0.696i)T - 2.80e3iT^{2} \)
59 \( 1 - 39.9iT - 3.48e3T^{2} \)
61 \( 1 - 5.90T + 3.72e3T^{2} \)
67 \( 1 + (45.1 + 45.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 68T + 5.04e3T^{2} \)
73 \( 1 + (-77.7 + 77.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (13.1 - 13.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 82.1iT - 7.92e3T^{2} \)
97 \( 1 + (24.5 + 24.5i)T + 9.40e3iT^{2} \)
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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

−15.30106042284178764033864070816, −14.38728672632753705309493049731, −13.32148296492164874895034386775, −11.92479787681270639481821757864, −10.41800818782164215387769935815, −9.655097719625892381817765838046, −8.188936133818958881658739208814, −6.06202788229051233680934759356, −5.10501206982701768295656437057, −1.94628610476765168941637162294, 2.87321794513497850062959245309, 5.10702721076156948153381292325, 7.02480997532644840892864870565, 8.114717944921096017346362135338, 9.700456788522033621998290250452, 10.89046501409081724666196908799, 12.37274349364423814275431497276, 13.43960148980225080769750743421, 14.41694277414791267581696338039, 15.93186082755117039812723930012

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