Properties

Label 2-3808-1.1-c1-0-84
Degree $2$
Conductor $3808$
Sign $-1$
Analytic cond. $30.4070$
Root an. cond. $5.51425$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.692·3-s + 2.18·5-s − 7-s − 2.52·9-s − 4.76·11-s + 3.00·13-s + 1.51·15-s + 17-s − 3.29·19-s − 0.692·21-s − 1.05·23-s − 0.225·25-s − 3.82·27-s + 3.24·29-s + 1.23·31-s − 3.29·33-s − 2.18·35-s + 1.17·37-s + 2.07·39-s − 1.21·41-s − 0.149·43-s − 5.50·45-s + 0.529·47-s + 49-s + 0.692·51-s − 6.32·53-s − 10.4·55-s + ⋯
L(s)  = 1  + 0.399·3-s + 0.977·5-s − 0.377·7-s − 0.840·9-s − 1.43·11-s + 0.833·13-s + 0.390·15-s + 0.242·17-s − 0.756·19-s − 0.151·21-s − 0.220·23-s − 0.0450·25-s − 0.735·27-s + 0.603·29-s + 0.221·31-s − 0.573·33-s − 0.369·35-s + 0.193·37-s + 0.333·39-s − 0.189·41-s − 0.0228·43-s − 0.821·45-s + 0.0773·47-s + 0.142·49-s + 0.0969·51-s − 0.868·53-s − 1.40·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3808\)    =    \(2^{5} \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(30.4070\)
Root analytic conductor: \(5.51425\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3808,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
17 \( 1 - T \)
good3 \( 1 - 0.692T + 3T^{2} \)
5 \( 1 - 2.18T + 5T^{2} \)
11 \( 1 + 4.76T + 11T^{2} \)
13 \( 1 - 3.00T + 13T^{2} \)
19 \( 1 + 3.29T + 19T^{2} \)
23 \( 1 + 1.05T + 23T^{2} \)
29 \( 1 - 3.24T + 29T^{2} \)
31 \( 1 - 1.23T + 31T^{2} \)
37 \( 1 - 1.17T + 37T^{2} \)
41 \( 1 + 1.21T + 41T^{2} \)
43 \( 1 + 0.149T + 43T^{2} \)
47 \( 1 - 0.529T + 47T^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 + 2.93T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 7.61T + 67T^{2} \)
71 \( 1 + 4.23T + 71T^{2} \)
73 \( 1 - 0.412T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 0.652T + 83T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 - 8.93T + 97T^{2} \)
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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

−8.220199823383204399226348795284, −7.56430243492982655588393743235, −6.41554977878388079724059633879, −5.92745801778416480448149729873, −5.31197308970410456953090090638, −4.29587508293990532033012130487, −3.12230710723398814600782280558, −2.62674609962138086070908280520, −1.63660433969957739827661779609, 0, 1.63660433969957739827661779609, 2.62674609962138086070908280520, 3.12230710723398814600782280558, 4.29587508293990532033012130487, 5.31197308970410456953090090638, 5.92745801778416480448149729873, 6.41554977878388079724059633879, 7.56430243492982655588393743235, 8.220199823383204399226348795284

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