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Properties

Label 2-1044-29.6-c1-0-3
Degree $2$
Conductor $1044$
Sign $0.0294 - 0.999i$
Analytic cond. $8.33638$
Root an. cond. $2.88727$
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  + (−1.39 + 1.74i)5-s + (0.343 − 1.50i)7-s + (0.891 + 1.85i)11-s + (1.60 − 0.773i)13-s + 2.10i·17-s + (−2.01 + 0.459i)19-s + (1.16 + 1.46i)23-s + (−0.00200 − 0.00880i)25-s + (0.599 + 5.35i)29-s + (−0.817 − 0.652i)31-s + (2.15 + 2.70i)35-s + (−1.92 + 4.00i)37-s + 3.55i·41-s + (−3.09 + 2.47i)43-s + (2.38 + 4.95i)47-s + ⋯
L(s)  = 1  + (−0.624 + 0.782i)5-s + (0.129 − 0.568i)7-s + (0.268 + 0.558i)11-s + (0.445 − 0.214i)13-s + 0.510i·17-s + (−0.461 + 0.105i)19-s + (0.243 + 0.305i)23-s + (−0.000401 − 0.00176i)25-s + (0.111 + 0.993i)29-s + (−0.146 − 0.117i)31-s + (0.364 + 0.456i)35-s + (−0.316 + 0.657i)37-s + 0.554i·41-s + (−0.472 + 0.376i)43-s + (0.348 + 0.723i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0294 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0294 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $0.0294 - 0.999i$
Analytic conductor: \(8.33638\)
Root analytic conductor: \(2.88727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :1/2),\ 0.0294 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.210800821\)
\(L(\frac12)\) \(\approx\) \(1.210800821\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
29 \( 1 + (-0.599 - 5.35i)T \)
good5 \( 1 + (1.39 - 1.74i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (-0.343 + 1.50i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-0.891 - 1.85i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.60 + 0.773i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 - 2.10iT - 17T^{2} \)
19 \( 1 + (2.01 - 0.459i)T + (17.1 - 8.24i)T^{2} \)
23 \( 1 + (-1.16 - 1.46i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (0.817 + 0.652i)T + (6.89 + 30.2i)T^{2} \)
37 \( 1 + (1.92 - 4.00i)T + (-23.0 - 28.9i)T^{2} \)
41 \( 1 - 3.55iT - 41T^{2} \)
43 \( 1 + (3.09 - 2.47i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-2.38 - 4.95i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (5.96 - 7.47i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 - 0.130T + 59T^{2} \)
61 \( 1 + (-0.310 - 0.0708i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (-2.40 - 1.15i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (0.643 - 0.309i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (3.78 - 3.01i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (-2.68 + 5.58i)T + (-49.2 - 61.7i)T^{2} \)
83 \( 1 + (-0.978 - 4.28i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (6.28 + 5.01i)T + (19.8 + 86.7i)T^{2} \)
97 \( 1 + (-8.86 + 2.02i)T + (87.3 - 42.0i)T^{2} \)
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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

−10.33431465855755961966823381110, −9.322598172597599062399231071574, −8.358958925185583136458069085225, −7.52676687804054604233215963104, −6.89928159866304378934090587338, −6.02653211560097113545245562863, −4.74617002624243161711714965924, −3.85414160684934146369945172065, −2.99849653739368790618394552884, −1.44980701085341555648670509966, 0.58135585914603728033637856371, 2.15483522985784501253109998382, 3.52188794123221625363383061296, 4.44237319705581076564867870655, 5.34843101485119342680739343382, 6.26619561271794028120825016559, 7.26234959974476687653318926252, 8.385901288700504186645350228230, 8.663185099549505123301268311708, 9.542719328999879870977252194880

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