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Properties

Label 2-45-9.2-c4-0-7
Degree $2$
Conductor $45$
Sign $-0.805 + 0.593i$
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  + (−5.11 − 2.95i)2-s + (1.28 + 8.90i)3-s + (9.41 + 16.3i)4-s + (−9.68 + 5.59i)5-s + (19.7 − 49.3i)6-s + (36.1 − 62.6i)7-s − 16.7i·8-s + (−77.7 + 22.8i)9-s + 65.9·10-s + (−187. − 108. i)11-s + (−133. + 104. i)12-s + (−13.6 − 23.6i)13-s + (−369. + 213. i)14-s + (−62.2 − 79.0i)15-s + (101. − 175. i)16-s − 339. i·17-s + ⋯
L(s)  = 1  + (−1.27 − 0.737i)2-s + (0.142 + 0.989i)3-s + (0.588 + 1.01i)4-s + (−0.387 + 0.223i)5-s + (0.548 − 1.36i)6-s + (0.737 − 1.27i)7-s − 0.261i·8-s + (−0.959 + 0.281i)9-s + 0.659·10-s + (−1.55 − 0.896i)11-s + (−0.925 + 0.727i)12-s + (−0.0808 − 0.140i)13-s + (−1.88 + 1.08i)14-s + (−0.276 − 0.351i)15-s + (0.395 − 0.685i)16-s − 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.593i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.805 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.805 + 0.593i$
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: \((2,\ 45,\ (\ :2),\ -0.805 + 0.593i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.106326 - 0.323627i\)
\(L(\frac12)\) \(\approx\) \(0.106326 - 0.323627i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.28 - 8.90i)T \)
5 \( 1 + (9.68 - 5.59i)T \)
good2 \( 1 + (5.11 + 2.95i)T + (8 + 13.8i)T^{2} \)
7 \( 1 + (-36.1 + 62.6i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (187. + 108. i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (13.6 + 23.6i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 339. iT - 8.35e4T^{2} \)
19 \( 1 + 301.T + 1.30e5T^{2} \)
23 \( 1 + (-329. + 190. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-136. - 78.9i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (201. + 348. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 291.T + 1.87e6T^{2} \)
41 \( 1 + (2.54e3 - 1.47e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.20e3 - 2.07e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-2.05e3 - 1.18e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 1.56e3iT - 7.89e6T^{2} \)
59 \( 1 + (-2.63e3 + 1.51e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (977. - 1.69e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (1.89e3 + 3.28e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 1.20e3iT - 2.54e7T^{2} \)
73 \( 1 + 6.11e3T + 2.83e7T^{2} \)
79 \( 1 + (-931. + 1.61e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-721. - 416. i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 9.60e3iT - 6.27e7T^{2} \)
97 \( 1 + (3.46e3 - 6.00e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−14.75795122060604920358096204133, −13.52871654487031555551272591286, −11.38550788631049605707302285861, −10.79197708333819447927190397679, −10.04235872072942208106835133410, −8.533976346241133951651813090954, −7.67117330883457323544373159754, −4.86803610415363232916164563438, −2.95563001276716921979149465173, −0.30823086928740305727598759657, 1.98179731264412840099010321227, 5.52404506260092901730823178322, 7.11754890201537126366511092572, 8.199825367140666921600035090004, 8.771895398384685775395249983823, 10.52771932703049184184332384440, 12.11204757552558569601074049200, 13.02272878837472029552740732474, 15.01731458352687971407121175733, 15.42253323231461449348878591647

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