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Properties

Label 2-45-15.8-c1-0-0
Degree $2$
Conductor $45$
Sign $0.927 - 0.374i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.870674 + 0.169098i\)
\(L(\frac12)\) \(\approx\) \(0.870674 + 0.169098i\)
\(L(\frac{3}{2})\) 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

−15.79566702023185403025722972810, −14.95368490624571016517932823633, −13.87713906290812859513187640333, −12.58162062090223483647164173253, −11.32065327001559831372721487565, −9.985583919322433016490765967424, −8.388180912004417505596354636959, −6.76997249157825406892103851238, −5.61024030463355534518954703534, −3.66957350149866266741830980981, 3.37217461998631009762624239828, 4.62890793123549973374705059312, 7.09039865126819512771646727198, 8.200621657208690564393827562023, 9.993211947347321831766557155915, 11.39070334637959316061896944773, 12.41304975759627672742515055192, 13.16671207025571892214490506474, 14.49497566323496116448276709853, 16.01977780028921854621463145392

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