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Properties

Label 2-45-9.2-c2-0-2
Degree $2$
Conductor $45$
Sign $0.503 - 0.863i$
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  + (2.67 + 1.54i)2-s + (−2.82 + 1.01i)3-s + (2.77 + 4.81i)4-s + (1.93 − 1.11i)5-s + (−9.12 − 1.63i)6-s + (1.10 − 1.91i)7-s + 4.80i·8-s + (6.92 − 5.75i)9-s + 6.91·10-s + (−15.4 − 8.91i)11-s + (−12.7 − 10.7i)12-s + (1.25 + 2.17i)13-s + (5.90 − 3.40i)14-s + (−4.32 + 5.12i)15-s + (3.67 − 6.37i)16-s + 32.6i·17-s + ⋯
L(s)  = 1  + (1.33 + 0.772i)2-s + (−0.940 + 0.339i)3-s + (0.694 + 1.20i)4-s + (0.387 − 0.223i)5-s + (−1.52 − 0.271i)6-s + (0.157 − 0.272i)7-s + 0.601i·8-s + (0.768 − 0.639i)9-s + 0.691·10-s + (−1.40 − 0.810i)11-s + (−1.06 − 0.895i)12-s + (0.0966 + 0.167i)13-s + (0.421 − 0.243i)14-s + (−0.288 + 0.341i)15-s + (0.229 − 0.398i)16-s + 1.91i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 - 0.863i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.503 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.503 - 0.863i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ 0.503 - 0.863i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.41991 + 0.815530i\)
\(L(\frac12)\) \(\approx\) \(1.41991 + 0.815530i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.82 - 1.01i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
good2 \( 1 + (-2.67 - 1.54i)T + (2 + 3.46i)T^{2} \)
7 \( 1 + (-1.10 + 1.91i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (15.4 + 8.91i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.25 - 2.17i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 32.6iT - 289T^{2} \)
19 \( 1 - 7.93T + 361T^{2} \)
23 \( 1 + (18.4 - 10.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (30.7 + 17.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (1.01 + 1.75i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 50.6T + 1.36e3T^{2} \)
41 \( 1 + (-4.65 + 2.68i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (7.76 - 13.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (1.63 + 0.943i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 62.0iT - 2.80e3T^{2} \)
59 \( 1 + (-31.3 + 18.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-21.1 + 36.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-7.38 - 12.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 105. iT - 5.04e3T^{2} \)
73 \( 1 + 66.9T + 5.32e3T^{2} \)
79 \( 1 + (-34.5 + 59.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (18.3 + 10.6i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 7.16iT - 7.92e3T^{2} \)
97 \( 1 + (55.6 - 96.4i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−15.77884581159971117684567587673, −14.74957078610601867628458891717, −13.37568899234343007320777561956, −12.75640587781508119013369375595, −11.29672262102674642454882479366, −10.05959159959998186197330961534, −7.85231836171629231033954253737, −6.15337996571347549921312785314, −5.43047685549586334019634748079, −3.97466547293388918834526802868, 2.44365620246364490109423337174, 4.81200728839911488711872357565, 5.68550673586613385615664426993, 7.40157530779351892139270726138, 9.994165907638620168331177867495, 11.08224229750760406665304887078, 12.05014282216676288101198703306, 12.98854858406818093748779932087, 13.82406422016912423779793286325, 15.19195411177087649619687342902

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