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Properties

Label 2-45-9.4-c3-0-6
Degree $2$
Conductor $45$
Sign $0.958 + 0.286i$
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  + (0.0874 + 0.151i)2-s + (5.19 − 0.151i)3-s + (3.98 − 6.90i)4-s + (−2.5 + 4.33i)5-s + (0.477 + 0.773i)6-s + (−4.23 − 7.32i)7-s + 2.79·8-s + (26.9 − 1.57i)9-s − 0.874·10-s + (15.7 + 27.2i)11-s + (19.6 − 36.4i)12-s + (−13.4 + 23.2i)13-s + (0.740 − 1.28i)14-s + (−12.3 + 22.8i)15-s + (−31.6 − 54.7i)16-s − 44.3·17-s + ⋯
L(s)  = 1  + (0.0309 + 0.0535i)2-s + (0.999 − 0.0291i)3-s + (0.498 − 0.862i)4-s + (−0.223 + 0.387i)5-s + (0.0324 + 0.0526i)6-s + (−0.228 − 0.395i)7-s + 0.123·8-s + (0.998 − 0.0583i)9-s − 0.0276·10-s + (0.431 + 0.747i)11-s + (0.472 − 0.876i)12-s + (−0.286 + 0.496i)13-s + (0.0141 − 0.0244i)14-s + (−0.212 + 0.393i)15-s + (−0.494 − 0.856i)16-s − 0.632·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.958 + 0.286i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.958 + 0.286i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.81721 - 0.266036i\)
\(L(\frac12)\) \(\approx\) \(1.81721 - 0.266036i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.19 + 0.151i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
good2 \( 1 + (-0.0874 - 0.151i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (4.23 + 7.32i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-15.7 - 27.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (13.4 - 23.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 44.3T + 4.91e3T^{2} \)
19 \( 1 + 90.2T + 6.85e3T^{2} \)
23 \( 1 + (97.1 - 168. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (1.87 + 3.24i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-125. + 217. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 62.2T + 5.06e4T^{2} \)
41 \( 1 + (-102. + 176. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-263. - 456. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (77.8 + 134. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 141.T + 1.48e5T^{2} \)
59 \( 1 + (-246. + 427. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-379. - 657. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-271. + 470. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 928.T + 3.57e5T^{2} \)
73 \( 1 - 608.T + 3.89e5T^{2} \)
79 \( 1 + (307. + 532. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (537. + 931. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.50e3T + 7.04e5T^{2} \)
97 \( 1 + (-166. - 288. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.11045838575390954301676108615, −14.36825821764204557617089207470, −13.27142800246579590024308655673, −11.66130927265110064576032108598, −10.26555658111203054151601414585, −9.350612851872587895812426769509, −7.56891031611451449660741584176, −6.49952833781939380312495125318, −4.20018360126835296418777212789, −2.09297552573748822880765211191, 2.58428612889672244095595164725, 4.11553204742542410785112687333, 6.63624926200621944600628098967, 8.182439607014758553411323289523, 8.842235804070830816872315385224, 10.60792448522987558285264519238, 12.19180970826160065294246237581, 12.93546138369724728053823854732, 14.23740486360877440491890479140, 15.52645127040185896794504904376

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