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Properties

Label 2-768-12.11-c1-0-7
Degree $2$
Conductor $768$
Sign $-i$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
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  + 1.73·3-s + 3.46i·5-s + 2i·7-s + 2.99·9-s − 3.46·11-s + 5.99i·15-s + 3.46i·21-s − 6.99·25-s + 5.19·27-s + 10.3i·29-s − 10i·31-s − 5.99·33-s − 6.92·35-s + 10.3i·45-s + 3·49-s + ⋯
L(s)  = 1  + 1.00·3-s + 1.54i·5-s + 0.755i·7-s + 0.999·9-s − 1.04·11-s + 1.54i·15-s + 0.755i·21-s − 1.39·25-s + 1.00·27-s + 1.92i·29-s − 1.79i·31-s − 1.04·33-s − 1.17·35-s + 1.54i·45-s + 0.428·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40258 + 1.40258i\)
\(L(\frac12)\) \(\approx\) \(1.40258 + 1.40258i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 10.3iT - 29T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 3.46iT - 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.48971901354522302832364767553, −9.715401122909150904043729428708, −8.822242900305511294082045724310, −7.87088267455179116597779560720, −7.23716426122048259564048818696, −6.32327614962902531203610541779, −5.19522761895446153394699703063, −3.74800611960441040979023191349, −2.83838686951482260099094827100, −2.19664772209953578996182997816, 0.912602202984102835619506873527, 2.27893758085900756219810726486, 3.69222820900469522900583935906, 4.58051703211354682183803819234, 5.36615790469085580561597214526, 6.83324376074904436367170150595, 7.909711051869976335881813279343, 8.277080756654716251741115385638, 9.194464113499630952432481745675, 9.930397457547826135757495616788

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