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Properties

Label 2-45-15.2-c1-0-0
Degree $2$
Conductor $45$
Sign $0.662 - 0.749i$
Analytic cond. $0.359326$
Root an. cond. $0.599438$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + 0.999i·4-s + (2.12 + 0.707i)5-s + (−2 − 2i)7-s + (−2.12 − 2.12i)8-s + (−2 + 0.999i)10-s − 2.82i·11-s + (1 − i)13-s + 2.82·14-s + 1.00·16-s + (−2.82 + 2.82i)17-s + (−0.707 + 2.12i)20-s + (2.00 + 2.00i)22-s + (2.82 + 2.82i)23-s + (3.99 + 3i)25-s + 1.41i·26-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + 0.499i·4-s + (0.948 + 0.316i)5-s + (−0.755 − 0.755i)7-s + (−0.750 − 0.750i)8-s + (−0.632 + 0.316i)10-s − 0.852i·11-s + (0.277 − 0.277i)13-s + 0.755·14-s + 0.250·16-s + (−0.685 + 0.685i)17-s + (−0.158 + 0.474i)20-s + (0.426 + 0.426i)22-s + (0.589 + 0.589i)23-s + (0.799 + 0.600i)25-s + 0.277i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.612410 + 0.276172i\)
\(L(\frac12)\) \(\approx\) \(0.612410 + 0.276172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.12 - 0.707i)T \)
good2 \( 1 + (0.707 - 0.707i)T - 2iT^{2} \)
7 \( 1 + (2 + 2i)T + 7iT^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (2.82 - 2.82i)T - 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-1 - i)T + 37iT^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + (8 - 8i)T - 43iT^{2} \)
47 \( 1 + (-5.65 + 5.65i)T - 47iT^{2} \)
53 \( 1 + (-2.82 - 2.82i)T + 53iT^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-4 - 4i)T + 67iT^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (-1 + i)T - 73iT^{2} \)
79 \( 1 + 12iT - 79T^{2} \)
83 \( 1 + (-2.82 - 2.82i)T + 83iT^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (11 + 11i)T + 97iT^{2} \)
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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

−16.39344105984239333031314046829, −15.08745333904504383550164131395, −13.52568710547677835615717632114, −12.95307518012515114317856213564, −11.07648150369747908571594140616, −9.813601344157596459911875113709, −8.674279076147765457799089192129, −7.13559581725033654267931037680, −6.05555100337708750065849810943, −3.42056620703233254502636074303, 2.25947599794642124177219078867, 5.26059532998057285416782806393, 6.56961946982146046531103976405, 8.989915349608132040349712194177, 9.523530201797702459658885758880, 10.73402037157714484917951818189, 12.16949045852122741856206588635, 13.30406841225363053987471170979, 14.61060898791995921357691401004, 15.71164015686282303751705000901

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