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Properties

Label 2-45-45.4-c1-0-3
Degree $2$
Conductor $45$
Sign $0.872 + 0.488i$
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  + (1.67 − 0.965i)2-s + (−1.67 + 0.448i)3-s + (0.866 − 1.50i)4-s + (−2.09 + 0.792i)5-s + (−2.36 + 2.36i)6-s + (0.776 − 0.448i)7-s + 0.517i·8-s + (2.59 − 1.50i)9-s + (−2.73 + 3.34i)10-s + (−2.36 − 4.09i)11-s + (−0.776 + 2.89i)12-s + (2.12 + 1.22i)13-s + (0.866 − 1.50i)14-s + (3.14 − 2.26i)15-s + (2.23 + 3.86i)16-s + 0.378i·17-s + ⋯
L(s)  = 1  + (1.18 − 0.683i)2-s + (−0.965 + 0.258i)3-s + (0.433 − 0.750i)4-s + (−0.935 + 0.354i)5-s + (−0.965 + 0.965i)6-s + (0.293 − 0.169i)7-s + 0.183i·8-s + (0.866 − 0.5i)9-s + (−0.863 + 1.05i)10-s + (−0.713 − 1.23i)11-s + (−0.224 + 0.836i)12-s + (0.588 + 0.339i)13-s + (0.231 − 0.400i)14-s + (0.811 − 0.584i)15-s + (0.558 + 0.966i)16-s + 0.0919i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.946520 - 0.246828i\)
\(L(\frac12)\) \(\approx\) \(0.946520 - 0.246828i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 - 0.448i)T \)
5 \( 1 + (2.09 - 0.792i)T \)
good2 \( 1 + (-1.67 + 0.965i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-0.776 + 0.448i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.36 + 4.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.12 - 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.378iT - 17T^{2} \)
19 \( 1 + 2.73T + 19T^{2} \)
23 \( 1 + (1.67 + 0.965i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.23 - 5.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.36 + 2.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.24iT - 37T^{2} \)
41 \( 1 + (2.13 - 3.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.91 - 4.57i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.79 + 2.19i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.86iT - 53T^{2} \)
59 \( 1 + (-1.26 + 2.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.33 + 9.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.45 - 2.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.80T + 71T^{2} \)
73 \( 1 + 8.48iT - 73T^{2} \)
79 \( 1 + (-0.267 - 0.464i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.02 + 5.20i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.39T + 89T^{2} \)
97 \( 1 + (8.90 - 5.13i)T + (48.5 - 84.0i)T^{2} \)
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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

−15.68936037661310093880467917980, −14.50407941054231237811638291061, −13.26059582549936840659233151371, −12.16043112807290934132212929560, −11.17761564294142646094874816322, −10.69007699270294443289187580493, −8.235601823355759550425838099252, −6.29060068472210823633506581082, −4.82087524989054237548181658641, −3.53658687071640279818941773544, 4.29990823860780370590205816847, 5.29860820502979910190626681338, 6.77799752033855071011296975596, 7.951088943996130534695981124934, 10.28785459524726241120347842202, 11.80808462310258572700290070629, 12.55854724526281259284739526759, 13.51112654662852440764628839805, 15.22658234956860481282968740204, 15.59736085223624226346254995482

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