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Properties

Label 2-45-45.29-c2-0-4
Degree $2$
Conductor $45$
Sign $0.985 - 0.170i$
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  + (−0.668 + 1.15i)2-s + (0.320 − 2.98i)3-s + (1.10 + 1.91i)4-s + (4.23 − 2.65i)5-s + (3.24 + 2.36i)6-s + (7.10 + 4.10i)7-s − 8.30·8-s + (−8.79 − 1.91i)9-s + (0.244 + 6.68i)10-s + (−5.67 − 3.27i)11-s + (6.06 − 2.68i)12-s + (1.29 − 0.749i)13-s + (−9.50 + 5.49i)14-s + (−6.56 − 13.4i)15-s + (1.13 − 1.97i)16-s − 15.1·17-s + ⋯
L(s)  = 1  + (−0.334 + 0.579i)2-s + (0.106 − 0.994i)3-s + (0.276 + 0.478i)4-s + (0.847 − 0.531i)5-s + (0.540 + 0.394i)6-s + (1.01 + 0.586i)7-s − 1.03·8-s + (−0.977 − 0.212i)9-s + (0.0244 + 0.668i)10-s + (−0.515 − 0.297i)11-s + (0.505 − 0.223i)12-s + (0.0998 − 0.0576i)13-s + (−0.679 + 0.392i)14-s + (−0.437 − 0.899i)15-s + (0.0710 − 0.123i)16-s − 0.889·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.985 - 0.170i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ 0.985 - 0.170i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.11913 + 0.0961186i\)
\(L(\frac12)\) \(\approx\) \(1.11913 + 0.0961186i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.320 + 2.98i)T \)
5 \( 1 + (-4.23 + 2.65i)T \)
good2 \( 1 + (0.668 - 1.15i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (-7.10 - 4.10i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (5.67 + 3.27i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.29 + 0.749i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 15.1T + 289T^{2} \)
19 \( 1 + 25.9T + 361T^{2} \)
23 \( 1 + (-11.6 - 20.1i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-6.96 - 4.02i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-22.5 - 38.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 62.8iT - 1.36e3T^{2} \)
41 \( 1 + (-9.97 + 5.75i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (36.9 + 21.3i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (8.25 - 14.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 66.0T + 2.80e3T^{2} \)
59 \( 1 + (-0.373 + 0.215i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-15.7 + 27.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-83.1 + 47.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 84.2iT - 5.04e3T^{2} \)
73 \( 1 - 63.5iT - 5.32e3T^{2} \)
79 \( 1 + (-9.06 + 15.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (50.4 - 87.4i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 86.3iT - 7.92e3T^{2} \)
97 \( 1 + (-59.7 - 34.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−15.73067331168679261760986062922, −14.50289162953085672990796069905, −13.23662424033831412231628614138, −12.34373893687300200877221275497, −11.10135446820483853864535770188, −8.838261861450142972421340906016, −8.284517163312024928323555714105, −6.78975721692745436079015503154, −5.49757900314930166945962436960, −2.24449268370324841970653219697, 2.38633511987613589269834583600, 4.73594006835469114535369407081, 6.37506745960650496821212747390, 8.550487729243069684932627728247, 9.945660293530081571326674118530, 10.63930771577132634094388927615, 11.40255605262774137508380999032, 13.46781628916707029068515572193, 14.73636073380954299757057639641, 15.20294583330812373831179851106

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