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Properties

Label 2-45-45.4-c3-0-7
Degree $2$
Conductor $45$
Sign $0.691 + 0.722i$
Analytic cond. $2.65508$
Root an. cond. $1.62944$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.31 + 1.33i)2-s + (−5.14 − 0.694i)3-s + (−0.422 + 0.730i)4-s + (7.31 − 8.45i)5-s + (12.8 − 5.27i)6-s + (13.3 − 7.70i)7-s − 23.6i·8-s + (26.0 + 7.14i)9-s + (−5.65 + 29.3i)10-s + (−22.2 − 38.5i)11-s + (2.68 − 3.47i)12-s + (24.2 + 14.0i)13-s + (−20.6 + 35.7i)14-s + (−43.5 + 38.4i)15-s + (28.2 + 48.9i)16-s − 92.6i·17-s + ⋯
L(s)  = 1  + (−0.819 + 0.472i)2-s + (−0.991 − 0.133i)3-s + (−0.0527 + 0.0913i)4-s + (0.654 − 0.755i)5-s + (0.874 − 0.359i)6-s + (0.720 − 0.416i)7-s − 1.04i·8-s + (0.964 + 0.264i)9-s + (−0.178 + 0.928i)10-s + (−0.610 − 1.05i)11-s + (0.0644 − 0.0835i)12-s + (0.517 + 0.298i)13-s + (−0.393 + 0.681i)14-s + (−0.749 + 0.661i)15-s + (0.441 + 0.765i)16-s − 1.32i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.691 + 0.722i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.591011 - 0.252395i\)
\(L(\frac12)\) \(\approx\) \(0.591011 - 0.252395i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.14 + 0.694i)T \)
5 \( 1 + (-7.31 + 8.45i)T \)
good2 \( 1 + (2.31 - 1.33i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (-13.3 + 7.70i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (22.2 + 38.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-24.2 - 14.0i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 92.6iT - 4.91e3T^{2} \)
19 \( 1 + 49.5T + 6.85e3T^{2} \)
23 \( 1 + (-0.799 - 0.461i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-94.9 - 164. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-149. + 259. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 57.8iT - 5.06e4T^{2} \)
41 \( 1 + (-143. + 249. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-0.512 + 0.295i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (518. - 299. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 146. iT - 1.48e5T^{2} \)
59 \( 1 + (96.5 - 167. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-283. - 490. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-307. - 177. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 320.T + 3.57e5T^{2} \)
73 \( 1 + 636. iT - 3.89e5T^{2} \)
79 \( 1 + (143. + 249. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-246. + 142. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 331.T + 7.04e5T^{2} \)
97 \( 1 + (-1.57e3 + 910. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−15.94736898815332409161398647269, −13.79301861922648166392848103902, −12.92285002875836424905524079321, −11.50167840599081197236033586517, −10.26318538439233747500486516256, −8.916460490669333936073038622016, −7.72199597588563919917766154407, −6.20092680556279746056386672031, −4.69604155941233244145284311078, −0.805992203644917574125858571709, 1.83760177078592943400179298693, 5.01514486510931856204011158329, 6.34218670766544085977821229320, 8.220259862749929515577195711855, 9.928424720107164854331720630673, 10.51981944556523612542965916220, 11.49742736870008801267386562479, 12.91535449624533842681742698293, 14.55473491114262376502604014619, 15.45507721401529344089645439093

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