Properties

Label 2-768-32.13-c1-0-8
Degree $2$
Conductor $768$
Sign $0.998 - 0.0605i$
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  + (−0.382 + 0.923i)3-s + (3.09 − 1.28i)5-s + (1.73 − 1.73i)7-s + (−0.707 − 0.707i)9-s + (2.39 + 5.79i)11-s + (−0.0173 − 0.00717i)13-s + 3.34i·15-s − 5.57i·17-s + (−1.03 − 0.426i)19-s + (0.938 + 2.26i)21-s + (−2.01 − 2.01i)23-s + (4.39 − 4.39i)25-s + (0.923 − 0.382i)27-s + (0.706 − 1.70i)29-s − 1.38·31-s + ⋯
L(s)  = 1  + (−0.220 + 0.533i)3-s + (1.38 − 0.572i)5-s + (0.655 − 0.655i)7-s + (−0.235 − 0.235i)9-s + (0.723 + 1.74i)11-s + (−0.00480 − 0.00199i)13-s + 0.864i·15-s − 1.35i·17-s + (−0.236 − 0.0979i)19-s + (0.204 + 0.494i)21-s + (−0.420 − 0.420i)23-s + (0.878 − 0.878i)25-s + (0.177 − 0.0736i)27-s + (0.131 − 0.316i)29-s − 0.247·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.998 - 0.0605i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.998 - 0.0605i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98132 + 0.0600428i\)
\(L(\frac12)\) \(\approx\) \(1.98132 + 0.0600428i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.382 - 0.923i)T \)
good5 \( 1 + (-3.09 + 1.28i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.73 + 1.73i)T - 7iT^{2} \)
11 \( 1 + (-2.39 - 5.79i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.0173 + 0.00717i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 5.57iT - 17T^{2} \)
19 \( 1 + (1.03 + 0.426i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.01 + 2.01i)T + 23iT^{2} \)
29 \( 1 + (-0.706 + 1.70i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 1.38T + 31T^{2} \)
37 \( 1 + (-2.87 + 1.19i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-6.97 - 6.97i)T + 41iT^{2} \)
43 \( 1 + (-1.67 - 4.03i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 1.15iT - 47T^{2} \)
53 \( 1 + (2.56 + 6.19i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (0.735 - 0.304i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (4.82 - 11.6i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (2.05 - 4.97i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-1.78 + 1.78i)T - 71iT^{2} \)
73 \( 1 + (-1.67 - 1.67i)T + 73iT^{2} \)
79 \( 1 - 2.67iT - 79T^{2} \)
83 \( 1 + (6.91 + 2.86i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-6.73 + 6.73i)T - 89iT^{2} \)
97 \( 1 + 1.75T + 97T^{2} \)
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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

−10.02702011164455166550659543296, −9.651031043165151201273594141802, −8.975872122115281993592816651170, −7.68350614564101260489548172250, −6.78265431366925784449143713134, −5.78935179133040840466844529319, −4.71815743235907823264178091407, −4.36975194045611029067879980485, −2.45837995264677848212891349381, −1.32063561411183292885272599310, 1.43276689342470872466722745162, 2.39332491657838923749411771856, 3.67598901687767383208415143899, 5.35341761881035736116364196440, 6.05335881914798952156094967778, 6.40645415063583079099682432952, 7.81000765352680231478015892587, 8.671377921407283916409819341106, 9.328398137446640205096701926789, 10.57115692494948402203158243140

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