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Properties

Label 2-45-45.14-c4-0-18
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: $\chi_{45} (14, \cdot )$
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

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

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