Properties

Label 2-8550-1.1-c1-0-46
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 − 2.43·7-s + 8-s + 4.08·11-s + 3.79·13-s − 2.43·14-s + 16-s − 3.73·17-s + 19-s + 4.08·22-s − 0.351·23-s + 3.79·26-s − 2.43·28-s + 6.73·29-s + 9.34·31-s + 32-s − 3.73·34-s − 6.17·37-s + 38-s + 4.05·41-s + 0.913·43-s + 4.08·44-s − 0.351·46-s + 5.52·47-s − 1.05·49-s + 3.79·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.921·7-s + 0.353·8-s + 1.23·11-s + 1.05·13-s − 0.651·14-s + 0.250·16-s − 0.905·17-s + 0.229·19-s + 0.871·22-s − 0.0733·23-s + 0.743·26-s − 0.460·28-s + 1.25·29-s + 1.67·31-s + 0.176·32-s − 0.640·34-s − 1.01·37-s + 0.162·38-s + 0.633·41-s + 0.139·43-s + 0.616·44-s − 0.0518·46-s + 0.805·47-s − 0.150·49-s + 0.525·52-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

\(L(1)\) \(\approx\) \(3.447048211\)
\(L(\frac12)\) \(\approx\) \(3.447048211\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.43T + 7T^{2} \)
11 \( 1 - 4.08T + 11T^{2} \)
13 \( 1 - 3.79T + 13T^{2} \)
17 \( 1 + 3.73T + 17T^{2} \)
23 \( 1 + 0.351T + 23T^{2} \)
29 \( 1 - 6.73T + 29T^{2} \)
31 \( 1 - 9.34T + 31T^{2} \)
37 \( 1 + 6.17T + 37T^{2} \)
41 \( 1 - 4.05T + 41T^{2} \)
43 \( 1 - 0.913T + 43T^{2} \)
47 \( 1 - 5.52T + 47T^{2} \)
53 \( 1 + 6.96T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 + 13.1T + 61T^{2} \)
67 \( 1 + 7.61T + 67T^{2} \)
71 \( 1 - 9.73T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 3.17T + 79T^{2} \)
83 \( 1 + 2.90T + 83T^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
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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

−7.66314433719462315108476863437, −6.71356411083531853867863444944, −6.35725201065930785945557507554, −6.00221017337040224056485351019, −4.77602086070496884053760557578, −4.30213120317229113320878845579, −3.44493295092591757962225673317, −2.95571960065060410089340804715, −1.82383135188140536533507806397, −0.837686227781796766820280731618, 0.837686227781796766820280731618, 1.82383135188140536533507806397, 2.95571960065060410089340804715, 3.44493295092591757962225673317, 4.30213120317229113320878845579, 4.77602086070496884053760557578, 6.00221017337040224056485351019, 6.35725201065930785945557507554, 6.71356411083531853867863444944, 7.66314433719462315108476863437

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