Properties

Label 2-2850-1.1-c1-0-19
Degree $2$
Conductor $2850$
Sign $1$
Analytic cond. $22.7573$
Root an. cond. $4.77046$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 4·7-s + 8-s + 9-s + 4·11-s + 12-s − 4·14-s + 16-s + 2·17-s + 18-s + 19-s − 4·21-s + 4·22-s + 2·23-s + 24-s + 27-s − 4·28-s − 6·29-s + 6·31-s + 32-s + 4·33-s + 2·34-s + 36-s + 8·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.872·21-s + 0.852·22-s + 0.417·23-s + 0.204·24-s + 0.192·27-s − 0.755·28-s − 1.11·29-s + 1.07·31-s + 0.176·32-s + 0.696·33-s + 0.342·34-s + 1/6·36-s + 1.31·37-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.414798048\)
\(L(\frac12)\) \(\approx\) \(3.414798048\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.075576842406541201241594770774, −7.80954843090816149956704777250, −7.22822931232102684143493050402, −6.24265288481434745554412487281, −6.04547848262920215925763806415, −4.68475865669594320244755229333, −3.85809577170289018122041589134, −3.25425866639495471915764029739, −2.44562386899404817546775712870, −1.04269711977505524501535875684, 1.04269711977505524501535875684, 2.44562386899404817546775712870, 3.25425866639495471915764029739, 3.85809577170289018122041589134, 4.68475865669594320244755229333, 6.04547848262920215925763806415, 6.24265288481434745554412487281, 7.22822931232102684143493050402, 7.80954843090816149956704777250, 9.075576842406541201241594770774

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