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Properties

Label 2-45-45.13-c2-0-2
Degree $2$
Conductor $45$
Sign $0.987 + 0.160i$
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  + (−3.30 − 0.884i)2-s + (−2.58 + 1.52i)3-s + (6.65 + 3.84i)4-s + (3.76 − 3.29i)5-s + (9.87 − 2.76i)6-s + (6.43 + 1.72i)7-s + (−8.91 − 8.91i)8-s + (4.32 − 7.89i)9-s + (−15.3 + 7.54i)10-s + (7.11 + 12.3i)11-s + (−23.0 + 0.259i)12-s + (6.99 − 1.87i)13-s + (−19.7 − 11.3i)14-s + (−4.67 + 14.2i)15-s + (6.17 + 10.6i)16-s + (−2.45 + 2.45i)17-s + ⋯
L(s)  = 1  + (−1.65 − 0.442i)2-s + (−0.860 + 0.509i)3-s + (1.66 + 0.960i)4-s + (0.752 − 0.658i)5-s + (1.64 − 0.460i)6-s + (0.919 + 0.246i)7-s + (−1.11 − 1.11i)8-s + (0.480 − 0.877i)9-s + (−1.53 + 0.754i)10-s + (0.646 + 1.11i)11-s + (−1.92 + 0.0215i)12-s + (0.538 − 0.144i)13-s + (−1.40 − 0.813i)14-s + (−0.311 + 0.950i)15-s + (0.385 + 0.668i)16-s + (−0.144 + 0.144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.987 + 0.160i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ 0.987 + 0.160i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.520455 - 0.0420463i\)
\(L(\frac12)\) \(\approx\) \(0.520455 - 0.0420463i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.58 - 1.52i)T \)
5 \( 1 + (-3.76 + 3.29i)T \)
good2 \( 1 + (3.30 + 0.884i)T + (3.46 + 2i)T^{2} \)
7 \( 1 + (-6.43 - 1.72i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-7.11 - 12.3i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-6.99 + 1.87i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (2.45 - 2.45i)T - 289iT^{2} \)
19 \( 1 + 10.6iT - 361T^{2} \)
23 \( 1 + (-30.9 + 8.28i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + (15.6 - 9.01i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (15.1 - 26.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (8.20 - 8.20i)T - 1.36e3iT^{2} \)
41 \( 1 + (-15.1 + 26.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (1.86 - 6.95i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (66.7 + 17.8i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (18.4 + 18.4i)T + 2.80e3iT^{2} \)
59 \( 1 + (5.75 + 3.32i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-35.6 - 61.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (10.0 + 37.5i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 25.2T + 5.04e3T^{2} \)
73 \( 1 + (14.2 + 14.2i)T + 5.32e3iT^{2} \)
79 \( 1 + (37.9 - 21.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (15.6 - 58.2i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 165. iT - 7.92e3T^{2} \)
97 \( 1 + (7.06 + 1.89i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−16.16317554184770481550713657998, −14.84558349417780181754095411757, −12.70389442922779072252957373803, −11.56099833046146940811004292215, −10.62523620031845028250196644854, −9.497454369127353966604736579856, −8.683554833256223217529111173238, −6.87481829957984662925296457017, −4.97503081773297589903739170861, −1.48601783483568918176368058184, 1.43101319241300988204583939630, 5.82696988743484530624648387026, 6.87925402081974197505869679398, 8.100297177902308443667987274436, 9.512282127750232726996929080046, 11.00973932896389707660079134684, 11.21466834683211762308137176803, 13.43794049387324927060074668752, 14.72442166206055512460361489571, 16.27711771958508173256947012198

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