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Properties

Label 2-45-45.34-c3-0-8
Degree $2$
Conductor $45$
Sign $0.910 - 0.414i$
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 + (3.65 + 10.5i)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 + (26.1 + 51.8i)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 + (0.327 + 0.944i)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 + (0.450 + 0.892i)15-s + (0.441 − 0.765i)16-s − 1.32i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.910 - 0.414i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.27088 + 0.492169i\)
\(L(\frac12)\) \(\approx\) \(2.27088 + 0.492169i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.65 - 10.5i)T \)
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

−15.16696155299614182344929454346, −14.18598631495343282129241884436, −13.55518010606497180648619244703, −12.45867187522019455872038255374, −10.21198123278799219337586668382, −9.572878526301884698890474876166, −7.38558563825097196189376033396, −6.57648874770262503882039231332, −4.53402655149008780369636883733, −2.81805851370199355496789679384, 2.57693183476727085237076932457, 4.06960338824606518836725598633, 5.66781962405065378191942466901, 8.148880982068988583220079495670, 8.940791892708632758925710029393, 10.44367608022238942166497778888, 12.33350351489473986980229382719, 13.01304163986596657612238667290, 13.74896659626862478033107583059, 15.02227214467098669864929053704

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