Properties

Label 2-384-32.13-c1-0-7
Degree $2$
Conductor $384$
Sign $-0.389 + 0.921i$
Analytic cond. $3.06625$
Root an. cond. $1.75107$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.389 + 0.921i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.389 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.509508 - 0.768582i\)
\(L(\frac12)\) \(\approx\) \(0.509508 - 0.768582i\)
\(L(\frac{3}{2})\) 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

−11.26030809825233100569002956349, −10.47574797638020698880308339079, −8.929531611027837492511924816442, −7.909712443630814484992903404728, −7.62648261961420774488193697809, −6.50537442252383629933466665747, −5.11407886828881831189538435637, −3.77155729200446084831510148838, −2.85413644875030989091990781261, −0.60104857931648503933516614187, 2.09589996828176496633301996811, 3.82932848638666798332073574714, 4.59454247129856311147919453613, 5.51499104683981402692963666700, 7.28269845819316015731518635406, 8.011650849806593412880133084510, 8.704340627500512713867939204567, 9.793324212561098975078394343353, 10.74097109940796084884123106754, 11.73091845527189517713131992153

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