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Properties

Label 2-1044-348.143-c0-0-0
Degree $2$
Conductor $1044$
Sign $0.893 - 0.448i$
Analytic cond. $0.521023$
Root an. cond. $0.721819$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.993 − 0.111i)2-s + (0.974 + 0.222i)4-s + (1.17 + 1.47i)5-s + (−0.943 − 0.330i)8-s + (−1.00 − 1.59i)10-s + (0.846 − 1.75i)13-s + (0.900 + 0.433i)16-s + (0.881 − 0.881i)17-s + (0.819 + 1.70i)20-s + (−0.570 + 2.49i)25-s + (−1.03 + 1.65i)26-s + (−0.943 + 0.330i)29-s + (−0.846 − 0.532i)32-s + (−0.974 + 0.777i)34-s + (−0.351 + 1.00i)37-s + ⋯
L(s)  = 1  + (−0.993 − 0.111i)2-s + (0.974 + 0.222i)4-s + (1.17 + 1.47i)5-s + (−0.943 − 0.330i)8-s + (−1.00 − 1.59i)10-s + (0.846 − 1.75i)13-s + (0.900 + 0.433i)16-s + (0.881 − 0.881i)17-s + (0.819 + 1.70i)20-s + (−0.570 + 2.49i)25-s + (−1.03 + 1.65i)26-s + (−0.943 + 0.330i)29-s + (−0.846 − 0.532i)32-s + (−0.974 + 0.777i)34-s + (−0.351 + 1.00i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $0.893 - 0.448i$
Analytic conductor: \(0.521023\)
Root analytic conductor: \(0.721819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :0),\ 0.893 - 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8567141446\)
\(L(\frac12)\) \(\approx\) \(0.8567141446\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.993 + 0.111i)T \)
3 \( 1 \)
29 \( 1 + (0.943 - 0.330i)T \)
good5 \( 1 + (-1.17 - 1.47i)T + (-0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.781 + 0.623i)T^{2} \)
13 \( 1 + (-0.846 + 1.75i)T + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.881 + 0.881i)T - iT^{2} \)
19 \( 1 + (-0.433 + 0.900i)T^{2} \)
23 \( 1 + (-0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.974 + 0.222i)T^{2} \)
37 \( 1 + (0.351 - 1.00i)T + (-0.781 - 0.623i)T^{2} \)
41 \( 1 + (0.314 + 0.314i)T + iT^{2} \)
43 \( 1 + (-0.974 + 0.222i)T^{2} \)
47 \( 1 + (0.781 - 0.623i)T^{2} \)
53 \( 1 + (0.175 - 0.139i)T + (0.222 - 0.974i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (1.05 - 1.68i)T + (-0.433 - 0.900i)T^{2} \)
67 \( 1 + (0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.0250 + 0.222i)T + (-0.974 + 0.222i)T^{2} \)
79 \( 1 + (-0.781 - 0.623i)T^{2} \)
83 \( 1 + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.862 + 0.0971i)T + (0.974 + 0.222i)T^{2} \)
97 \( 1 + (0.900 + 1.43i)T + (-0.433 + 0.900i)T^{2} \)
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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

−10.07803291031921973307087739166, −9.676161647254830858241357807622, −8.594118085919061821712786012525, −7.62304700744686348333731154828, −7.00553759187606618670653277232, −6.00755965585054119723637608697, −5.53773720333944895209655825645, −3.25202849435789442039743542161, −2.90029632976825192873415695146, −1.52464147002597321171327736407, 1.40110944501895667713955294983, 1.99919882411820552426231712986, 3.84906273099564609852320610302, 5.11115820453378197955233086379, 5.97949866139726882252608817598, 6.57857165161577870060231994133, 7.88195657343668967129742178899, 8.624029898524926036818334432787, 9.275042251228899987768139771796, 9.684472692154569526904013975931

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