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Properties

Label 2-45-9.2-c4-0-13
Degree $2$
Conductor $45$
Sign $0.0766 + 0.997i$
Analytic cond. $4.65164$
Root an. cond. $2.15676$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.10 + 1.79i)2-s + (−5.44 − 7.16i)3-s + (−1.56 − 2.71i)4-s + (−9.68 + 5.59i)5-s + (−4.04 − 32.0i)6-s + (45.9 − 79.5i)7-s − 68.6i·8-s + (−21.7 + 78.0i)9-s − 40.0·10-s + (−110. − 63.8i)11-s + (−10.9 + 26.0i)12-s + (37.9 + 65.7i)13-s + (285. − 164. i)14-s + (92.7 + 38.9i)15-s + (97.9 − 169. i)16-s + 325. i·17-s + ⋯
L(s)  = 1  + (0.776 + 0.448i)2-s + (−0.604 − 0.796i)3-s + (−0.0981 − 0.169i)4-s + (−0.387 + 0.223i)5-s + (−0.112 − 0.889i)6-s + (0.936 − 1.62i)7-s − 1.07i·8-s + (−0.269 + 0.963i)9-s − 0.400·10-s + (−0.914 − 0.527i)11-s + (−0.0760 + 0.180i)12-s + (0.224 + 0.389i)13-s + (1.45 − 0.839i)14-s + (0.412 + 0.173i)15-s + (0.382 − 0.662i)16-s + 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0766 + 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0766 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.0766 + 0.997i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.0766 + 0.997i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.13965 - 1.05543i\)
\(L(\frac12)\) \(\approx\) \(1.13965 - 1.05543i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.44 + 7.16i)T \)
5 \( 1 + (9.68 - 5.59i)T \)
good2 \( 1 + (-3.10 - 1.79i)T + (8 + 13.8i)T^{2} \)
7 \( 1 + (-45.9 + 79.5i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (110. + 63.8i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-37.9 - 65.7i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 325. iT - 8.35e4T^{2} \)
19 \( 1 - 384.T + 1.30e5T^{2} \)
23 \( 1 + (-125. + 72.4i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-738. - 426. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (734. + 1.27e3i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 398.T + 1.87e6T^{2} \)
41 \( 1 + (-2.02e3 + 1.17e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-364. + 630. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (103. + 59.9i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 2.47e3iT - 7.89e6T^{2} \)
59 \( 1 + (-894. + 516. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (246. - 427. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.99e3 - 6.92e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 1.32e3iT - 2.54e7T^{2} \)
73 \( 1 - 7.30e3T + 2.83e7T^{2} \)
79 \( 1 + (3.40e3 - 5.90e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (6.00e3 + 3.46e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 1.20e3iT - 6.27e7T^{2} \)
97 \( 1 + (476. - 825. i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−14.43261756169063536918607681854, −13.73587959934270625592725714646, −12.83947614972191927758879512184, −11.23066878880061255760280545443, −10.44107631700070039475491430633, −7.933216637228638829943286955367, −7.00228428268697878993439411702, −5.54286731911860372494219518354, −4.16010559682300681574058014072, −0.906392919396130633537773133718, 2.89182489404578051603535975291, 4.83693554899040827295333610898, 5.39334505711773963742121459121, 8.050585864386523589209836336501, 9.299260178182221463324403783735, 11.09549298310815044290807255568, 11.87216329031961527357147082482, 12.63496373596539280366926937753, 14.29398474218908696683207747605, 15.36388679778378169095411786334

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