Properties

Label 2-9200-1.1-c1-0-31
Degree $2$
Conductor $9200$
Sign $1$
Analytic cond. $73.4623$
Root an. cond. $8.57101$
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  + 0.794·3-s − 2.47·7-s − 2.36·9-s + 2.29·11-s − 3.84·13-s − 7.74·17-s + 2.29·19-s − 1.96·21-s + 23-s − 4.26·27-s + 5.28·29-s + 6.40·31-s + 1.82·33-s + 8.56·37-s − 3.05·39-s + 4.27·41-s + 1.88·43-s − 12.3·47-s − 0.889·49-s − 6.15·51-s − 7.57·53-s + 1.82·57-s − 6.07·59-s − 0.635·61-s + 5.85·63-s − 11.1·67-s + 0.794·69-s + ⋯
L(s)  = 1  + 0.458·3-s − 0.934·7-s − 0.789·9-s + 0.693·11-s − 1.06·13-s − 1.87·17-s + 0.527·19-s − 0.428·21-s + 0.208·23-s − 0.821·27-s + 0.981·29-s + 1.14·31-s + 0.318·33-s + 1.40·37-s − 0.488·39-s + 0.667·41-s + 0.288·43-s − 1.80·47-s − 0.127·49-s − 0.862·51-s − 1.04·53-s + 0.242·57-s − 0.790·59-s − 0.0813·61-s + 0.737·63-s − 1.36·67-s + 0.0956·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9200\)    =    \(2^{4} \cdot 5^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(73.4623\)
Root analytic conductor: \(8.57101\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.334053077\)
\(L(\frac12)\) \(\approx\) \(1.334053077\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 - T \)
good3 \( 1 - 0.794T + 3T^{2} \)
7 \( 1 + 2.47T + 7T^{2} \)
11 \( 1 - 2.29T + 11T^{2} \)
13 \( 1 + 3.84T + 13T^{2} \)
17 \( 1 + 7.74T + 17T^{2} \)
19 \( 1 - 2.29T + 19T^{2} \)
29 \( 1 - 5.28T + 29T^{2} \)
31 \( 1 - 6.40T + 31T^{2} \)
37 \( 1 - 8.56T + 37T^{2} \)
41 \( 1 - 4.27T + 41T^{2} \)
43 \( 1 - 1.88T + 43T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 + 7.57T + 53T^{2} \)
59 \( 1 + 6.07T + 59T^{2} \)
61 \( 1 + 0.635T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 + 8.58T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 + 0.335T + 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 5.55T + 89T^{2} \)
97 \( 1 - 6.42T + 97T^{2} \)
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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

−7.82028329168418627104335784057, −6.88318721580808481909673077400, −6.45634676705848745210127003612, −5.85751271913915969991706230203, −4.70233230127745056286249820324, −4.36137971196189676668339012571, −3.11453755756342998430908540303, −2.85562946277512344164100208726, −1.92294759158612784604942017186, −0.51403247337798881626771358821, 0.51403247337798881626771358821, 1.92294759158612784604942017186, 2.85562946277512344164100208726, 3.11453755756342998430908540303, 4.36137971196189676668339012571, 4.70233230127745056286249820324, 5.85751271913915969991706230203, 6.45634676705848745210127003612, 6.88318721580808481909673077400, 7.82028329168418627104335784057

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