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Properties

Label 2-45-45.14-c2-0-0
Degree $2$
Conductor $45$
Sign $-0.360 - 0.932i$
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  + (−1.84 − 3.19i)2-s + (−1.66 + 2.49i)3-s + (−4.79 + 8.31i)4-s + (−4.90 − 0.992i)5-s + (11.0 + 0.716i)6-s + (−2.22 + 1.28i)7-s + 20.6·8-s + (−3.45 − 8.31i)9-s + (5.86 + 17.4i)10-s + (−8.51 + 4.91i)11-s + (−12.7 − 25.8i)12-s + (−10.4 − 6.03i)13-s + (8.19 + 4.73i)14-s + (10.6 − 10.5i)15-s + (−18.8 − 32.6i)16-s − 4.28·17-s + ⋯
L(s)  = 1  + (−0.921 − 1.59i)2-s + (−0.555 + 0.831i)3-s + (−1.19 + 2.07i)4-s + (−0.980 − 0.198i)5-s + (1.83 + 0.119i)6-s + (−0.317 + 0.183i)7-s + 2.57·8-s + (−0.383 − 0.923i)9-s + (0.586 + 1.74i)10-s + (−0.773 + 0.446i)11-s + (−1.06 − 2.15i)12-s + (−0.803 − 0.463i)13-s + (0.585 + 0.338i)14-s + (0.709 − 0.705i)15-s + (−1.17 − 2.04i)16-s − 0.252·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.360 - 0.932i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ -0.360 - 0.932i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0198118 + 0.0288973i\)
\(L(\frac12)\) \(\approx\) \(0.0198118 + 0.0288973i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.66 - 2.49i)T \)
5 \( 1 + (4.90 + 0.992i)T \)
good2 \( 1 + (1.84 + 3.19i)T + (-2 + 3.46i)T^{2} \)
7 \( 1 + (2.22 - 1.28i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (8.51 - 4.91i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (10.4 + 6.03i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 4.28T + 289T^{2} \)
19 \( 1 - 7.16T + 361T^{2} \)
23 \( 1 + (-0.255 + 0.442i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (26.4 - 15.2i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-9.61 + 16.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 1.31iT - 1.36e3T^{2} \)
41 \( 1 + (29.9 + 17.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (44.9 - 25.9i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-25.4 - 44.1i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 86.6T + 2.80e3T^{2} \)
59 \( 1 + (-91.7 - 52.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (15.6 + 27.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (67.5 + 39.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 72.6iT - 5.04e3T^{2} \)
73 \( 1 + 30.3iT - 5.32e3T^{2} \)
79 \( 1 + (57.6 + 99.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-30.0 - 51.9i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 71.2iT - 7.92e3T^{2} \)
97 \( 1 + (-110. + 63.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−16.25062799511070514001255550096, −15.12784592550457582954612710951, −12.87251151214547499159129694925, −12.06270548073205353570064565626, −11.11353542328787051209790213345, −10.14024640813649190794469838177, −9.138792689783520551379682025401, −7.74278384821495654863692736002, −4.71778657159648273273069157997, −3.19199639644797582965238804190, 0.05190500612656212842672773216, 5.16821653702372764700367261668, 6.67873116546810763132471275895, 7.50146646007144031510523842809, 8.475356535003840345030699364863, 10.18379585064292388109044987412, 11.58747930978095281145458189737, 13.22022301754160620664391339833, 14.46195500848951655814426398582, 15.65085438731874350549264859528

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