Properties

Label 2-8550-1.1-c1-0-1
Degree $2$
Conductor $8550$
Sign $1$
Analytic cond. $68.2720$
Root an. cond. $8.26269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2.69·7-s − 8-s − 5.54·11-s − 2.91·13-s + 2.69·14-s + 16-s − 4.91·17-s + 19-s + 5.54·22-s − 3.60·23-s + 2.91·26-s − 2.69·28-s − 1.08·29-s + 7.54·31-s − 32-s + 4.91·34-s + 4.54·37-s − 38-s − 9.54·43-s − 5.54·44-s + 3.60·46-s + 0.836·47-s + 0.245·49-s − 2.91·52-s − 9.78·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.01·7-s − 0.353·8-s − 1.67·11-s − 0.809·13-s + 0.719·14-s + 0.250·16-s − 1.19·17-s + 0.229·19-s + 1.18·22-s − 0.752·23-s + 0.572·26-s − 0.508·28-s − 0.200·29-s + 1.35·31-s − 0.176·32-s + 0.843·34-s + 0.747·37-s − 0.162·38-s − 1.45·43-s − 0.836·44-s + 0.532·46-s + 0.122·47-s + 0.0350·49-s − 0.404·52-s − 1.34·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8550\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(68.2720\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2169151082\)
\(L(\frac12)\) \(\approx\) \(0.2169151082\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good7 \( 1 + 2.69T + 7T^{2} \)
11 \( 1 + 5.54T + 11T^{2} \)
13 \( 1 + 2.91T + 13T^{2} \)
17 \( 1 + 4.91T + 17T^{2} \)
23 \( 1 + 3.60T + 23T^{2} \)
29 \( 1 + 1.08T + 29T^{2} \)
31 \( 1 - 7.54T + 31T^{2} \)
37 \( 1 - 4.54T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 9.54T + 43T^{2} \)
47 \( 1 - 0.836T + 47T^{2} \)
53 \( 1 + 9.78T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 7.38T + 61T^{2} \)
67 \( 1 - 2.85T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 - 5.15T + 73T^{2} \)
79 \( 1 + 3.09T + 79T^{2} \)
83 \( 1 + 1.71T + 83T^{2} \)
89 \( 1 - 5.09T + 89T^{2} \)
97 \( 1 + 17.2T + 97T^{2} \)
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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

−7.933205542071202177401316833786, −7.16164643299102277471802509098, −6.49546702654402310999984706128, −5.91989154704053771698240690137, −5.00184010162464656513853689742, −4.34745191904989399585305937749, −3.04932136079510884204164235486, −2.73467663422044861728572007819, −1.78326952862109232266535767008, −0.23599053022095932960090973057, 0.23599053022095932960090973057, 1.78326952862109232266535767008, 2.73467663422044861728572007819, 3.04932136079510884204164235486, 4.34745191904989399585305937749, 5.00184010162464656513853689742, 5.91989154704053771698240690137, 6.49546702654402310999984706128, 7.16164643299102277471802509098, 7.933205542071202177401316833786

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