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Properties

Label 2-768-48.29-c0-0-3
Degree $2$
Conductor $768$
Sign $-0.382 + 0.923i$
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  + (−0.707 − 0.707i)3-s − 1.41i·7-s + 1.00i·9-s + (−1 − i)13-s + (−1.00 + 1.00i)21-s − i·25-s + (0.707 − 0.707i)27-s − 1.41·31-s + (1 − i)37-s + 1.41i·39-s − 1.00·49-s + (1 + i)61-s + 1.41·63-s + (1.41 + 1.41i)67-s + (−0.707 + 0.707i)75-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s − 1.41i·7-s + 1.00i·9-s + (−1 − i)13-s + (−1.00 + 1.00i)21-s − i·25-s + (0.707 − 0.707i)27-s − 1.41·31-s + (1 − i)37-s + 1.41i·39-s − 1.00·49-s + (1 + i)61-s + 1.41·63-s + (1.41 + 1.41i)67-s + (−0.707 + 0.707i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(0.383281\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6551410767\)
\(L(\frac12)\) \(\approx\) \(0.6551410767\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + iT^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.45654648858001826076431688013, −9.674456822114586874000020282693, −8.235966008383804841800828033247, −7.47448400418578472126877030440, −6.97786142857710253982331009585, −5.86771939584615238294315056329, −4.95346695150744839011667156222, −3.90513823797067780262373371644, −2.38076276436362608287248420143, −0.75137044797525364260919094343, 2.10110209145205183716226362850, 3.43294835580117660338348996374, 4.70122051681237414997937803449, 5.37356384890399184672899423801, 6.24251051548040817548455498901, 7.17887634069304071626328509295, 8.466911060945603222420061089713, 9.441288557782877988434647510542, 9.626310323968048748989983933636, 10.97906695086689203669690864401

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