Properties

Label 2-768-3.2-c0-0-0
Degree $2$
Conductor $768$
Sign $-i$
Analytic cond. $0.383281$
Root an. cond. $0.619097$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s − 9-s + 2i·11-s + 25-s i·27-s − 2·33-s − 49-s − 2i·59-s + 2·73-s + i·75-s + 81-s − 2i·83-s + 2·97-s − 2i·99-s − 2i·107-s + ⋯
L(s)  = 1  + i·3-s − 9-s + 2i·11-s + 25-s i·27-s − 2·33-s − 49-s − 2i·59-s + 2·73-s + i·75-s + 81-s − 2i·83-s + 2·97-s − 2i·99-s − 2i·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-i$
Analytic conductor: \(0.383281\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9082539329\)
\(L(\frac12)\) \(\approx\) \(0.9082539329\)
\(L(1)\) 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 \( 1 - iT \)
good5 \( 1 - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - 2iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 2iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−10.57579604182470278989178742714, −9.842610997450187228092650115146, −9.309281712207652705462010343223, −8.296824494016569845828844685206, −7.29631093228283246496110921738, −6.35265223941422036836609645592, −5.04220861762749904743178176906, −4.56634549720434133804464441339, −3.41007626517704594129377904426, −2.11411251405953769947321261844, 1.02777572148403383408704263892, 2.63984763760398080177827851398, 3.56565119294518014715373598869, 5.18517096887695521932682548074, 6.04541178656737113000583711432, 6.74423643649784553539034866123, 7.83229911155073064036663204386, 8.503231277316214331094544444154, 9.183908354581915427311547196559, 10.58411748633036289322809683421

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