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Properties

Label 2-45-45.29-c4-0-4
Degree $2$
Conductor $45$
Sign $-0.972 - 0.232i$
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  + (−2.61 + 4.53i)2-s + (8.42 + 3.16i)3-s + (−5.69 − 9.86i)4-s + (−19.7 + 15.3i)5-s + (−36.4 + 29.8i)6-s + (50.0 + 28.9i)7-s − 24.1·8-s + (60.9 + 53.3i)9-s + (−17.6 − 129. i)10-s + (−146. − 84.5i)11-s + (−16.7 − 101. i)12-s + (−99.1 + 57.2i)13-s + (−262. + 151. i)14-s + (−214. + 66.3i)15-s + (154. − 267. i)16-s + 209.·17-s + ⋯
L(s)  = 1  + (−0.654 + 1.13i)2-s + (0.935 + 0.352i)3-s + (−0.356 − 0.616i)4-s + (−0.790 + 0.612i)5-s + (−1.01 + 0.830i)6-s + (1.02 + 0.589i)7-s − 0.376·8-s + (0.752 + 0.659i)9-s + (−0.176 − 1.29i)10-s + (−1.21 − 0.699i)11-s + (−0.116 − 0.702i)12-s + (−0.586 + 0.338i)13-s + (−1.33 + 0.771i)14-s + (−0.955 + 0.295i)15-s + (0.602 − 1.04i)16-s + 0.726·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.972 - 0.232i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ -0.972 - 0.232i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.142490 + 1.20789i\)
\(L(\frac12)\) \(\approx\) \(0.142490 + 1.20789i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.42 - 3.16i)T \)
5 \( 1 + (19.7 - 15.3i)T \)
good2 \( 1 + (2.61 - 4.53i)T + (-8 - 13.8i)T^{2} \)
7 \( 1 + (-50.0 - 28.9i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (146. + 84.5i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (99.1 - 57.2i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 209.T + 8.35e4T^{2} \)
19 \( 1 - 26.9T + 1.30e5T^{2} \)
23 \( 1 + (-458. - 794. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-308. - 178. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-629. - 1.08e3i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 368. iT - 1.87e6T^{2} \)
41 \( 1 + (-1.59e3 + 918. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (277. + 160. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.22e3 + 2.12e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 1.89e3T + 7.89e6T^{2} \)
59 \( 1 + (4.20e3 - 2.42e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (473. - 820. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (1.87e3 - 1.08e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 9.07e3iT - 2.54e7T^{2} \)
73 \( 1 - 984. iT - 2.83e7T^{2} \)
79 \( 1 + (-5.12e3 + 8.87e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (2.39e3 - 4.14e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 858. iT - 6.27e7T^{2} \)
97 \( 1 + (-3.69e3 - 2.13e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−15.48734458707681059128566068563, −14.92606487304408386178576450363, −13.92493757415372531293780845813, −11.99647176143468823092883921552, −10.55695823713523330773919521277, −8.984464550943167001416209645676, −8.002310841848287942086503929056, −7.35262560698786453794300787075, −5.23051705490187172422545112470, −3.00247955464005453128824861899, 0.910989505282559059594941714639, 2.64049395090025666160656062663, 4.53157328173170824085533804778, 7.62211651497559526179774906546, 8.312157870736608567672603352153, 9.733231596851911538914375845114, 10.84950683540670592291160961350, 12.19612432209792304736461914396, 12.95320775702993920452587494144, 14.58168019293333353349625947079

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