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Properties

Label 2-45-9.5-c4-0-3
Degree $2$
Conductor $45$
Sign $-0.165 - 0.986i$
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  + (−6.31 + 3.64i)2-s + (−7.77 − 4.53i)3-s + (18.5 − 32.2i)4-s + (9.68 + 5.59i)5-s + (65.6 + 0.260i)6-s + (−25.8 − 44.8i)7-s + 154. i·8-s + (39.9 + 70.4i)9-s − 81.5·10-s + (−33.9 + 19.6i)11-s + (−290. + 166. i)12-s + (−83.2 + 144. i)13-s + (326. + 188. i)14-s + (−49.9 − 87.3i)15-s + (−265. − 460. i)16-s + 267. i·17-s + ⋯
L(s)  = 1  + (−1.57 + 0.911i)2-s + (−0.864 − 0.503i)3-s + (1.16 − 2.01i)4-s + (0.387 + 0.223i)5-s + (1.82 + 0.00723i)6-s + (−0.528 − 0.914i)7-s + 2.41i·8-s + (0.493 + 0.869i)9-s − 0.815·10-s + (−0.280 + 0.162i)11-s + (−2.01 + 1.15i)12-s + (−0.492 + 0.853i)13-s + (1.66 + 0.963i)14-s + (−0.222 − 0.388i)15-s + (−1.03 − 1.79i)16-s + 0.926i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.165 - 0.986i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ -0.165 - 0.986i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.263547 + 0.311563i\)
\(L(\frac12)\) \(\approx\) \(0.263547 + 0.311563i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (7.77 + 4.53i)T \)
5 \( 1 + (-9.68 - 5.59i)T \)
good2 \( 1 + (6.31 - 3.64i)T + (8 - 13.8i)T^{2} \)
7 \( 1 + (25.8 + 44.8i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (33.9 - 19.6i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (83.2 - 144. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 267. iT - 8.35e4T^{2} \)
19 \( 1 - 662.T + 1.30e5T^{2} \)
23 \( 1 + (-554. - 320. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (1.09e3 - 630. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (513. - 888. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 1.74e3T + 1.87e6T^{2} \)
41 \( 1 + (193. + 111. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-821. - 1.42e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-882. + 509. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 1.08e3iT - 7.89e6T^{2} \)
59 \( 1 + (1.55e3 + 898. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-2.25e3 - 3.90e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (595. - 1.03e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 7.19e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.39e3T + 2.83e7T^{2} \)
79 \( 1 + (4.05e3 + 7.01e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (2.36e3 - 1.36e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 2.91e3iT - 6.27e7T^{2} \)
97 \( 1 + (-4.34e3 - 7.52e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−16.07152653329991061924902020197, −14.55775399727855371251666423773, −13.15464512531982891721877929914, −11.33195587741039122380047174406, −10.31311234587818782273528218789, −9.361397218993054114046292688517, −7.47596090673215486882513780129, −6.92791445802585142116235648235, −5.55669284510786168992984444032, −1.28232390322337230579487315330, 0.54169557890380899747074005621, 2.86676403771284929778464768467, 5.56373750509497527701716951172, 7.47738648868826249581958867636, 9.272716397445405715664150934323, 9.697448454242542433006620172387, 11.00570923686137594703264770359, 11.93661635766694994659724120776, 12.90596074909112021166461060480, 15.40210240124280802464695019485

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