Properties

Label 2-1305-145.144-c1-0-1
Degree $2$
Conductor $1305$
Sign $-1$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
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  − 2·4-s + 2.23i·5-s + 5.38i·11-s + 4·16-s − 4.47i·20-s − 2.23i·23-s − 5.00·25-s − 5.38i·29-s − 12.0·37-s + 5.38i·41-s − 12.0·43-s − 10.7i·44-s + 7·49-s + 11.1i·53-s − 12.0·55-s + ⋯
L(s)  = 1  − 4-s + 0.999i·5-s + 1.62i·11-s + 16-s − 0.999i·20-s − 0.466i·23-s − 1.00·25-s − 0.999i·29-s − 1.97·37-s + 0.841i·41-s − 1.83·43-s − 1.62i·44-s + 49-s + 1.53i·53-s − 1.62·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $-1$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4996418101\)
\(L(\frac12)\) \(\approx\) \(0.4996418101\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23iT \)
29 \( 1 + 5.38iT \)
good2 \( 1 + 2T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 5.38iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 2.23iT - 23T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 12.0T + 37T^{2} \)
41 \( 1 - 5.38iT - 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 11.1iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 15.6iT - 83T^{2} \)
89 \( 1 + 10.7iT - 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−10.13910534994676241493609506508, −9.364219667902822261002809409808, −8.461168466785031876375582784570, −7.54338755701619960080542557189, −6.89969363321014212174241808316, −5.89743988967450134656050730817, −4.83225155921523667657761638756, −4.12907271947267493859959729610, −3.06491727752146568228042806351, −1.82680416816571196489098261312, 0.22269172729142053465133409986, 1.43833832894140756409834202688, 3.30538821934600939207968873120, 3.99274686870874226041305619069, 5.31977653041580870695401398586, 5.38568888522453096885479601980, 6.72479145089049766712017799806, 7.973200032362912146603035930454, 8.640541769201483444791419317800, 8.951859567458182178738431853278

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