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Properties

Label 2-1456-13.12-c1-0-17
Degree $2$
Conductor $1456$
Sign $0.0862 - 0.996i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.21·3-s + 3.21i·5-s − i·7-s + 1.90·9-s + 2.68i·11-s + (3.59 + 0.311i)13-s + 7.11i·15-s − 3.59·17-s + 8.54i·19-s − 2.21i·21-s − 3.28·23-s − 5.33·25-s − 2.42·27-s + 2.05·29-s − 5.83i·31-s + ⋯
L(s)  = 1  + 1.27·3-s + 1.43i·5-s − 0.377i·7-s + 0.634·9-s + 0.810i·11-s + (0.996 + 0.0862i)13-s + 1.83i·15-s − 0.871·17-s + 1.96i·19-s − 0.483i·21-s − 0.684·23-s − 1.06·25-s − 0.467·27-s + 0.380·29-s − 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0862 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0862 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.0862 - 0.996i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.0862 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.458460072\)
\(L(\frac12)\) \(\approx\) \(2.458460072\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
13 \( 1 + (-3.59 - 0.311i)T \)
good3 \( 1 - 2.21T + 3T^{2} \)
5 \( 1 - 3.21iT - 5T^{2} \)
11 \( 1 - 2.68iT - 11T^{2} \)
17 \( 1 + 3.59T + 17T^{2} \)
19 \( 1 - 8.54iT - 19T^{2} \)
23 \( 1 + 3.28T + 23T^{2} \)
29 \( 1 - 2.05T + 29T^{2} \)
31 \( 1 + 5.83iT - 31T^{2} \)
37 \( 1 + 3.93iT - 37T^{2} \)
41 \( 1 - 0.755iT - 41T^{2} \)
43 \( 1 - 8.80T + 43T^{2} \)
47 \( 1 - 1.88iT - 47T^{2} \)
53 \( 1 - 2.52T + 53T^{2} \)
59 \( 1 - 7.33iT - 59T^{2} \)
61 \( 1 - 9.05T + 61T^{2} \)
67 \( 1 + 0.428iT - 67T^{2} \)
71 \( 1 + 8.98iT - 71T^{2} \)
73 \( 1 + 5.79iT - 73T^{2} \)
79 \( 1 - 4.47T + 79T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 + 5.36iT - 89T^{2} \)
97 \( 1 - 9.62iT - 97T^{2} \)
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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

−9.733100360354159350588155524920, −8.886258227389658819069736926370, −7.931691365157578609554651788153, −7.53363311045592207596592644569, −6.55607586007700855672658719899, −5.86704458711096578636620185023, −4.08609641837020435039079321723, −3.71889384751308007101716524734, −2.62233052967325773937958133898, −1.86447380574767829043218562326, 0.839734517815158978252223269257, 2.17092339867793364235131585471, 3.15327515591736018569444226146, 4.16621783832338551552234980201, 5.01546477205294392771521698608, 5.96905572292917494067598098813, 7.02670414360429332442965333892, 8.229604622737745402067897298534, 8.676687357156461332666943597938, 8.899899195712092950269195684396

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