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Properties

Label 2-45-45.29-c2-0-9
Degree $2$
Conductor $45$
Sign $-0.323 + 0.946i$
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  + (1.42 − 2.46i)2-s + (−2.93 − 0.600i)3-s + (−2.03 − 3.52i)4-s + (−1.22 − 4.84i)5-s + (−5.65 + 6.38i)6-s + (8.42 + 4.86i)7-s − 0.212·8-s + (8.27 + 3.52i)9-s + (−13.6 − 3.86i)10-s + (0.370 + 0.214i)11-s + (3.87 + 11.5i)12-s + (−16.2 + 9.37i)13-s + (23.9 − 13.8i)14-s + (0.699 + 14.9i)15-s + (7.84 − 13.5i)16-s + 2.85·17-s + ⋯
L(s)  = 1  + (0.710 − 1.23i)2-s + (−0.979 − 0.200i)3-s + (−0.509 − 0.882i)4-s + (−0.245 − 0.969i)5-s + (−0.942 + 1.06i)6-s + (1.20 + 0.694i)7-s − 0.0265·8-s + (0.919 + 0.392i)9-s + (−1.36 − 0.386i)10-s + (0.0337 + 0.0194i)11-s + (0.322 + 0.966i)12-s + (−1.24 + 0.721i)13-s + (1.71 − 0.987i)14-s + (0.0466 + 0.998i)15-s + (0.490 − 0.849i)16-s + 0.168·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.323 + 0.946i$
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: \((2,\ 45,\ (\ :1),\ -0.323 + 0.946i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.722557 - 1.01054i\)
\(L(\frac12)\) \(\approx\) \(0.722557 - 1.01054i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.93 + 0.600i)T \)
5 \( 1 + (1.22 + 4.84i)T \)
good2 \( 1 + (-1.42 + 2.46i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (-8.42 - 4.86i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.370 - 0.214i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (16.2 - 9.37i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 2.85T + 289T^{2} \)
19 \( 1 - 0.530T + 361T^{2} \)
23 \( 1 + (-10.8 - 18.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (21.0 + 12.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-6.33 - 10.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 14.5iT - 1.36e3T^{2} \)
41 \( 1 + (33.1 - 19.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-50.0 - 28.8i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-24.7 + 42.9i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 44.5T + 2.80e3T^{2} \)
59 \( 1 + (54.6 - 31.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-11.0 + 19.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-28.1 + 16.2i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 89.8iT - 5.04e3T^{2} \)
73 \( 1 + 144. iT - 5.32e3T^{2} \)
79 \( 1 + (25.1 - 43.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (38.2 - 66.2i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 28.9iT - 7.92e3T^{2} \)
97 \( 1 + (19.9 + 11.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−15.09576007077532144942714226402, −13.65661089265436239457624173887, −12.41499335599166921673671531108, −11.89210089071430461012811047182, −11.11160779198708662999508654892, −9.538721071456780718359021624598, −7.68258158791819052917467481239, −5.29402815599841560261780101566, −4.54361019321349271869525705168, −1.71882273436169518746021168365, 4.33784355411549907123355882987, 5.50251954158190101671967630348, 6.98408816220448237993903007103, 7.71253813276943717923981368522, 10.32362089026539581458260822155, 11.16761319062702660415221710747, 12.61474165400103908936117694290, 14.18786608562937353152796735577, 14.82812764572506947219174858661, 15.75416323922052672430770210643

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