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Properties

Label 2-825-55.54-c2-0-61
Degree $2$
Conductor $825$
Sign $0.161 + 0.986i$
Analytic cond. $22.4796$
Root an. cond. $4.74126$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.35·2-s + 1.73i·3-s + 1.53·4-s + 4.07i·6-s − 6.42·7-s − 5.79·8-s − 2.99·9-s + (3.26 − 10.5i)11-s + 2.66i·12-s + 7.68·13-s − 15.1·14-s − 19.7·16-s + 8.15·17-s − 7.05·18-s − 30.4i·19-s + ⋯
L(s)  = 1  + 1.17·2-s + 0.577i·3-s + 0.383·4-s + 0.679i·6-s − 0.918·7-s − 0.724·8-s − 0.333·9-s + (0.297 − 0.954i)11-s + 0.221i·12-s + 0.591·13-s − 1.08·14-s − 1.23·16-s + 0.479·17-s − 0.392·18-s − 1.60i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.161 + 0.986i$
Analytic conductor: \(22.4796\)
Root analytic conductor: \(4.74126\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1),\ 0.161 + 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.890377365\)
\(L(\frac12)\) \(\approx\) \(1.890377365\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
5 \( 1 \)
11 \( 1 + (-3.26 + 10.5i)T \)
good2 \( 1 - 2.35T + 4T^{2} \)
7 \( 1 + 6.42T + 49T^{2} \)
13 \( 1 - 7.68T + 169T^{2} \)
17 \( 1 - 8.15T + 289T^{2} \)
19 \( 1 + 30.4iT - 361T^{2} \)
23 \( 1 + 31.6iT - 529T^{2} \)
29 \( 1 - 6.88iT - 841T^{2} \)
31 \( 1 - 51.1T + 961T^{2} \)
37 \( 1 - 19.4iT - 1.36e3T^{2} \)
41 \( 1 + 47.9iT - 1.68e3T^{2} \)
43 \( 1 + 81.8T + 1.84e3T^{2} \)
47 \( 1 + 30.1iT - 2.20e3T^{2} \)
53 \( 1 - 26.0iT - 2.80e3T^{2} \)
59 \( 1 + 82.7T + 3.48e3T^{2} \)
61 \( 1 + 75.4iT - 3.72e3T^{2} \)
67 \( 1 - 34iT - 4.48e3T^{2} \)
71 \( 1 + 72.7T + 5.04e3T^{2} \)
73 \( 1 - 54.8T + 5.32e3T^{2} \)
79 \( 1 - 24.3iT - 6.24e3T^{2} \)
83 \( 1 - 0.923T + 6.88e3T^{2} \)
89 \( 1 + 44.8T + 7.92e3T^{2} \)
97 \( 1 - 21.2iT - 9.40e3T^{2} \)
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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

−9.879961805461761980513997874137, −8.964625365235626545709181559195, −8.405957290960562350926540130524, −6.66862589048834094300584837630, −6.27735447699628078763967396242, −5.21499507098616068580958392760, −4.40833735398219060646882626141, −3.38638694402186271955960974073, −2.83373507663556692658547598292, −0.42648413184385779367089645420, 1.54321745607103305930250826589, 3.03765686614352284075997226973, 3.75332230209041089257256637853, 4.83339748008696943710181872572, 5.97683487938385396093107649922, 6.35890724284001835498404511365, 7.44701100534141527123991929900, 8.397762765531320726812767216366, 9.560438199114060771204030959959, 10.06025520142869752925996857542

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