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Properties

Label 2-1512-24.11-c1-0-54
Degree $2$
Conductor $1512$
Sign $-0.707 + 0.707i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + (−1.36 − 0.366i)2-s + (1.73 + i)4-s − 3.12·5-s − i·7-s + (−1.99 − 2i)8-s + (4.27 + 1.14i)10-s + 0.397i·11-s + 6.55i·13-s + (−0.366 + 1.36i)14-s + (1.99 + 3.46i)16-s − 2i·17-s − 0.527·19-s + (−5.42 − 3.12i)20-s + (0.145 − 0.543i)22-s + 8.21·23-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s − 1.39·5-s − 0.377i·7-s + (−0.707 − 0.707i)8-s + (1.35 + 0.362i)10-s + 0.119i·11-s + 1.81i·13-s + (−0.0978 + 0.365i)14-s + (0.499 + 0.866i)16-s − 0.485i·17-s − 0.121·19-s + (−1.21 − 0.699i)20-s + (0.0310 − 0.115i)22-s + 1.71·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2896289343\)
\(L(\frac12)\) \(\approx\) \(0.2896289343\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 3.12T + 5T^{2} \)
11 \( 1 - 0.397iT - 11T^{2} \)
13 \( 1 - 6.55iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 0.527T + 19T^{2} \)
23 \( 1 - 8.21T + 23T^{2} \)
29 \( 1 + 4.73T + 29T^{2} \)
31 \( 1 + 7.55iT - 31T^{2} \)
37 \( 1 + 3.35iT - 37T^{2} \)
41 \( 1 - 7.48iT - 41T^{2} \)
43 \( 1 - 1.62T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 - 0.795T + 53T^{2} \)
59 \( 1 - 2.47iT - 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 + 6.55T + 67T^{2} \)
71 \( 1 + 2.21T + 71T^{2} \)
73 \( 1 + 5.75T + 73T^{2} \)
79 \( 1 + 10.2iT - 79T^{2} \)
83 \( 1 + 8.36iT - 83T^{2} \)
89 \( 1 + 5.31iT - 89T^{2} \)
97 \( 1 + 19.1T + 97T^{2} \)
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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

−9.194355072304195903319505550059, −8.434323192728075300426157534806, −7.50681002034384858081178024193, −7.16050392185197908000681769196, −6.30276805761195072497438433653, −4.68021119601046419315326919754, −3.95066586670325070193973524198, −2.99698224409194797630532426197, −1.63977919637158342466306862780, −0.18652625230188282050883066096, 1.10433200883185721856661276180, 2.81647653705923047026546317141, 3.53352953118510871126761390099, 4.99656784855028509505381386251, 5.73208045099425389047980842457, 6.90470695452247998280508811365, 7.49476813359056727655151949442, 8.323307292642434861079640667717, 8.633103254993960327116906307497, 9.686416400896943077550074132350

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