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Properties

Label 2-45-9.4-c1-0-1
Degree $2$
Conductor $45$
Sign $0.586 - 0.809i$
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  + (1.04 + 1.80i)2-s + (−1.04 − 1.38i)3-s + (−1.17 + 2.03i)4-s + (−0.5 + 0.866i)5-s + (1.41 − 3.32i)6-s + (−2.04 − 3.53i)7-s − 0.734·8-s + (−0.824 + 2.88i)9-s − 2.08·10-s + (0.675 + 1.17i)11-s + (4.04 − 0.498i)12-s + (−0.324 + 0.561i)13-s + (4.26 − 7.38i)14-s + (1.71 − 0.211i)15-s + (1.58 + 2.74i)16-s − 1.35·17-s + ⋯
L(s)  = 1  + (0.737 + 1.27i)2-s + (−0.602 − 0.798i)3-s + (−0.587 + 1.01i)4-s + (−0.223 + 0.387i)5-s + (0.575 − 1.35i)6-s + (−0.772 − 1.33i)7-s − 0.259·8-s + (−0.274 + 0.961i)9-s − 0.659·10-s + (0.203 + 0.353i)11-s + (1.16 − 0.143i)12-s + (−0.0898 + 0.155i)13-s + (1.13 − 1.97i)14-s + (0.443 − 0.0547i)15-s + (0.396 + 0.686i)16-s − 0.327·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.586 - 0.809i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ 0.586 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802763 + 0.409533i\)
\(L(\frac12)\) \(\approx\) \(0.802763 + 0.409533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.04 + 1.38i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.04 - 1.80i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (2.04 + 3.53i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.675 - 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.324 - 0.561i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.35T + 17T^{2} \)
19 \( 1 - 0.648T + 19T^{2} \)
23 \( 1 + (2.39 - 4.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.93 + 3.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.84 + 6.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.52T + 37T^{2} \)
41 \( 1 + (-0.0898 + 0.155i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.410 - 0.710i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.45 + 9.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.17T + 53T^{2} \)
59 \( 1 + (2.08 - 3.61i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.91 - 3.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.07 - 7.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.11T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + (5.17 + 8.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.12 - 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-6.79 - 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−16.16411806774599775847909098682, −14.88630021574237651119322952539, −13.61889279812603858364531905490, −13.18616213902954527432610978770, −11.60867463198518106702045057164, −10.16218166475136889031121479286, −7.74022067356809975746706903007, −7.00572775022621672484051852226, −6.01865674004262494123430804984, −4.21214934995389426819785277996, 3.09312176866925448606064717414, 4.67761242782936460247511586809, 6.00939549294963706675553880929, 8.884947726371900626927213454153, 10.00807396555845522052903944309, 11.25263784495139753445860362544, 12.15809529035548401330996626524, 12.84287240466090350039328936299, 14.47351145043800891660113714211, 15.72480935386663217562861737340

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