Properties

Label 2-6900-1.1-c1-0-37
Degree $2$
Conductor $6900$
Sign $1$
Analytic cond. $55.0967$
Root an. cond. $7.42272$
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  − 3-s + 3.73·7-s + 9-s + 4.84·11-s + 2.84·13-s − 0.890·17-s + 6.84·19-s − 3.73·21-s − 23-s − 27-s − 0.890·29-s + 7.73·31-s − 4.84·33-s + 1.95·37-s − 2.84·39-s + 12.3·41-s + 3.47·43-s + 6.62·47-s + 6.95·49-s + 0.890·51-s − 12.3·53-s − 6.84·57-s + 0.890·59-s + 8.62·61-s + 3.73·63-s − 7.73·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.41·7-s + 0.333·9-s + 1.46·11-s + 0.788·13-s − 0.216·17-s + 1.57·19-s − 0.815·21-s − 0.208·23-s − 0.192·27-s − 0.165·29-s + 1.38·31-s − 0.843·33-s + 0.321·37-s − 0.455·39-s + 1.93·41-s + 0.529·43-s + 0.966·47-s + 0.993·49-s + 0.124·51-s − 1.69·53-s − 0.906·57-s + 0.115·59-s + 1.10·61-s + 0.470·63-s − 0.945·67-s + 0.120·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(55.0967\)
Root analytic conductor: \(7.42272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.710576125\)
\(L(\frac12)\) \(\approx\) \(2.710576125\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
23 \( 1 + T \)
good7 \( 1 - 3.73T + 7T^{2} \)
11 \( 1 - 4.84T + 11T^{2} \)
13 \( 1 - 2.84T + 13T^{2} \)
17 \( 1 + 0.890T + 17T^{2} \)
19 \( 1 - 6.84T + 19T^{2} \)
29 \( 1 + 0.890T + 29T^{2} \)
31 \( 1 - 7.73T + 31T^{2} \)
37 \( 1 - 1.95T + 37T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 - 3.47T + 43T^{2} \)
47 \( 1 - 6.62T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 - 0.890T + 59T^{2} \)
61 \( 1 - 8.62T + 61T^{2} \)
67 \( 1 + 7.73T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + 16.5T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 4.58T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
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

−7.78120433094349601966403777122, −7.40914190383440101794203270977, −6.38100131094481760597121226636, −5.94086566936277682712746730939, −5.09466904605026794345198319381, −4.38442564878572338036617375729, −3.84448019219689876184288175883, −2.67432596828391511103625222988, −1.40955959955311236775891446189, −1.05867100212817633948096638123, 1.05867100212817633948096638123, 1.40955959955311236775891446189, 2.67432596828391511103625222988, 3.84448019219689876184288175883, 4.38442564878572338036617375729, 5.09466904605026794345198319381, 5.94086566936277682712746730939, 6.38100131094481760597121226636, 7.40914190383440101794203270977, 7.78120433094349601966403777122

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