Properties

Label 2-2850-1.1-c1-0-9
Degree $2$
Conductor $2850$
Sign $1$
Analytic cond. $22.7573$
Root an. cond. $4.77046$
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 − 3-s + 4-s − 6-s − 4.42·7-s + 8-s + 9-s + 2.62·11-s − 12-s + 5.80·13-s − 4.42·14-s + 16-s − 3.80·17-s + 18-s + 19-s + 4.42·21-s + 2.62·22-s − 2.62·23-s − 24-s + 5.80·26-s − 27-s − 4.42·28-s + 3.37·29-s − 4.42·31-s + 32-s − 2.62·33-s − 3.80·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.67·7-s + 0.353·8-s + 0.333·9-s + 0.790·11-s − 0.288·12-s + 1.61·13-s − 1.18·14-s + 0.250·16-s − 0.923·17-s + 0.235·18-s + 0.229·19-s + 0.966·21-s + 0.559·22-s − 0.546·23-s − 0.204·24-s + 1.13·26-s − 0.192·27-s − 0.836·28-s + 0.627·29-s − 0.795·31-s + 0.176·32-s − 0.456·33-s − 0.652·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2850\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(22.7573\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.050977775\)
\(L(\frac12)\) \(\approx\) \(2.050977775\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
19 \( 1 - T \)
good7 \( 1 + 4.42T + 7T^{2} \)
11 \( 1 - 2.62T + 11T^{2} \)
13 \( 1 - 5.80T + 13T^{2} \)
17 \( 1 + 3.80T + 17T^{2} \)
23 \( 1 + 2.62T + 23T^{2} \)
29 \( 1 - 3.37T + 29T^{2} \)
31 \( 1 + 4.42T + 31T^{2} \)
37 \( 1 - 5.80T + 37T^{2} \)
41 \( 1 - 5.67T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 - 2.62T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 1.05T + 59T^{2} \)
61 \( 1 - 4.75T + 61T^{2} \)
67 \( 1 - 15.6T + 67T^{2} \)
71 \( 1 - 15.6T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 - 4.42T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 - 7.37T + 97T^{2} \)
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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

−8.935146476314441049796576933717, −7.919861130128087638081263057618, −6.70050469182400468847362832832, −6.47473306744498430484430295554, −5.95156157584620285987864921908, −4.90876453670553712726802206299, −3.75761581483371301093945822392, −3.56872708393186299936335304581, −2.21991208521503445339808477558, −0.819481846713935859873666670801, 0.819481846713935859873666670801, 2.21991208521503445339808477558, 3.56872708393186299936335304581, 3.75761581483371301093945822392, 4.90876453670553712726802206299, 5.95156157584620285987864921908, 6.47473306744498430484430295554, 6.70050469182400468847362832832, 7.919861130128087638081263057618, 8.935146476314441049796576933717

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