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Properties

Label 2-45-45.14-c2-0-2
Degree $2$
Conductor $45$
Sign $0.0797 - 0.996i$
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  + (1.14 + 1.98i)2-s + (−0.182 + 2.99i)3-s + (−0.629 + 1.09i)4-s + (0.662 − 4.95i)5-s + (−6.15 + 3.07i)6-s + (−6.04 + 3.49i)7-s + 6.28·8-s + (−8.93 − 1.09i)9-s + (10.6 − 4.36i)10-s + (15.8 − 9.12i)11-s + (−3.15 − 2.08i)12-s + (−3.66 − 2.11i)13-s + (−13.8 − 8.00i)14-s + (14.7 + 2.88i)15-s + (9.72 + 16.8i)16-s − 17.3·17-s + ⋯
L(s)  = 1  + (0.573 + 0.993i)2-s + (−0.0607 + 0.998i)3-s + (−0.157 + 0.272i)4-s + (0.132 − 0.991i)5-s + (−1.02 + 0.511i)6-s + (−0.863 + 0.498i)7-s + 0.785·8-s + (−0.992 − 0.121i)9-s + (1.06 − 0.436i)10-s + (1.43 − 0.829i)11-s + (−0.262 − 0.173i)12-s + (−0.282 − 0.162i)13-s + (−0.990 − 0.571i)14-s + (0.981 + 0.192i)15-s + (0.607 + 1.05i)16-s − 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0797 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0797 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.0797 - 0.996i$
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: \((2,\ 45,\ (\ :1),\ 0.0797 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.03825 + 0.958541i\)
\(L(\frac12)\) \(\approx\) \(1.03825 + 0.958541i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.182 - 2.99i)T \)
5 \( 1 + (-0.662 + 4.95i)T \)
good2 \( 1 + (-1.14 - 1.98i)T + (-2 + 3.46i)T^{2} \)
7 \( 1 + (6.04 - 3.49i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-15.8 + 9.12i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (3.66 + 2.11i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 17.3T + 289T^{2} \)
19 \( 1 + 3.96T + 361T^{2} \)
23 \( 1 + (-0.287 + 0.498i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (18.1 - 10.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (16.7 - 28.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 21.4iT - 1.36e3T^{2} \)
41 \( 1 + (-44.1 - 25.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (7.15 - 4.13i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (7.57 + 13.1i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 24.5T + 2.80e3T^{2} \)
59 \( 1 + (-43.1 - 24.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (31.4 + 54.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-103. - 59.7i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 66.8iT - 5.04e3T^{2} \)
73 \( 1 - 48.9iT - 5.32e3T^{2} \)
79 \( 1 + (-58.9 - 102. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-3.66 - 6.34i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 100. iT - 7.92e3T^{2} \)
97 \( 1 + (3.59 - 2.07i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−16.00512457137201257526767432009, −14.92939774509834665609479529192, −13.91441762562648086121266426401, −12.65569011986903993669024678010, −11.15626058296340910717159468686, −9.514069041515805454718697090701, −8.641846356701150683447544518937, −6.46214329264563744390051514449, −5.42289191754944835637019237260, −4.02694104350437408879623101922, 2.20289567176270860102901922860, 3.85005594779752640872282870343, 6.50164658672333376597082494318, 7.32877492292275247049451840253, 9.581696439629295290409110338640, 10.98739583367702718775557401462, 11.88434090759310527072376920214, 12.94675891028570479088422486010, 13.79819686660702695683996649980, 14.83536749203996603097053328467

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