Properties

Label 2-1512-8.5-c1-0-5
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 − i)2-s + 2i·4-s + 2i·5-s − 7-s + (2 − 2i)8-s + (2 − 2i)10-s + 5i·13-s + (1 + i)14-s − 4·16-s + 17-s + 4i·19-s − 4·20-s − 5·23-s + 25-s + (5 − 5i)26-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + i·4-s + 0.894i·5-s − 0.377·7-s + (0.707 − 0.707i)8-s + (0.632 − 0.632i)10-s + 1.38i·13-s + (0.267 + 0.267i)14-s − 16-s + 0.242·17-s + 0.917i·19-s − 0.894·20-s − 1.04·23-s + 0.200·25-s + (0.980 − 0.980i)26-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: $\chi_{1512} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4606952248\)
\(L(\frac12)\) \(\approx\) \(0.4606952248\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 2iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 5T + 23T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 - iT - 59T^{2} \)
61 \( 1 - 6iT - 61T^{2} \)
67 \( 1 + 9iT - 67T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 13T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.858836300135979305976447501723, −9.192392812184509574549061191963, −8.239869611956688114807726572980, −7.46850419789980652659799474827, −6.71349829997619331367582564672, −5.93579477733754355163933939893, −4.33117788465248853942571840832, −3.65446722905062973768644923531, −2.60911414813695091113651228143, −1.67577352415054073113134653995, 0.23429795009191015451726674852, 1.43751376113978214378912348920, 2.93910021077498731887237177472, 4.32943916376496872714672702190, 5.37029345297608884360089259703, 5.74872591673051753796580018429, 6.96582062291966178022602929668, 7.57497846125810977950257510947, 8.547536848463692056793550593836, 8.935168772701952214420735604333

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