Properties

Label 2-1248-1.1-c1-0-16
Degree $2$
Conductor $1248$
Sign $-1$
Analytic cond. $9.96533$
Root an. cond. $3.15679$
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  − 3-s − 1.23·5-s + 1.23·7-s + 9-s − 2·11-s − 13-s + 1.23·15-s + 4.47·17-s − 5.23·19-s − 1.23·21-s + 2.47·23-s − 3.47·25-s − 27-s + 4.47·29-s − 7.70·31-s + 2·33-s − 1.52·35-s + 0.472·37-s + 39-s + 1.23·41-s − 6.47·43-s − 1.23·45-s − 6.94·47-s − 5.47·49-s − 4.47·51-s + 8.47·53-s + 2.47·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.552·5-s + 0.467·7-s + 0.333·9-s − 0.603·11-s − 0.277·13-s + 0.319·15-s + 1.08·17-s − 1.20·19-s − 0.269·21-s + 0.515·23-s − 0.694·25-s − 0.192·27-s + 0.830·29-s − 1.38·31-s + 0.348·33-s − 0.258·35-s + 0.0776·37-s + 0.160·39-s + 0.193·41-s − 0.986·43-s − 0.184·45-s − 1.01·47-s − 0.781·49-s − 0.626·51-s + 1.16·53-s + 0.333·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-1$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
13 \( 1 + T \)
good5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 + 5.23T + 19T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 7.70T + 31T^{2} \)
37 \( 1 - 0.472T + 37T^{2} \)
41 \( 1 - 1.23T + 41T^{2} \)
43 \( 1 + 6.47T + 43T^{2} \)
47 \( 1 + 6.94T + 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 + 8.47T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 6.76T + 67T^{2} \)
71 \( 1 + 4.47T + 71T^{2} \)
73 \( 1 - 0.472T + 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 + 7.52T + 83T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 + 4.47T + 97T^{2} \)
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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

−9.351705555426971137426416943951, −8.275798701814878111155644201629, −7.72832915348017176959791716160, −6.85762564308701428781618770273, −5.84678043219083801586685780798, −5.03228506978679632896220437989, −4.21578031527768778010874229372, −3.07633394664636135572320304998, −1.65256455659439457675396194219, 0, 1.65256455659439457675396194219, 3.07633394664636135572320304998, 4.21578031527768778010874229372, 5.03228506978679632896220437989, 5.84678043219083801586685780798, 6.85762564308701428781618770273, 7.72832915348017176959791716160, 8.275798701814878111155644201629, 9.351705555426971137426416943951

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