Properties

Label 2-384-3.2-c4-0-52
Degree $2$
Conductor $384$
Sign $-0.929 + 0.369i$
Analytic cond. $39.6940$
Root an. cond. $6.30032$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.32 − 8.36i)3-s − 25.4i·5-s + 36.1·7-s + (−58.9 + 55.5i)9-s − 188. i·11-s + 223.·13-s + (−212. + 84.5i)15-s − 271. i·17-s + 255.·19-s + (−119. − 302. i)21-s − 308. i·23-s − 23.4·25-s + (660. + 308. i)27-s − 1.04e3i·29-s + 1.13e3·31-s + ⋯
L(s)  = 1  + (−0.369 − 0.929i)3-s − 1.01i·5-s + 0.737·7-s + (−0.727 + 0.685i)9-s − 1.55i·11-s + 1.32·13-s + (−0.946 + 0.375i)15-s − 0.938i·17-s + 0.708·19-s + (−0.272 − 0.685i)21-s − 0.582i·23-s − 0.0374·25-s + (0.906 + 0.423i)27-s − 1.24i·29-s + 1.17·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.929 + 0.369i$
Analytic conductor: \(39.6940\)
Root analytic conductor: \(6.30032\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :2),\ -0.929 + 0.369i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.907270939\)
\(L(\frac12)\) \(\approx\) \(1.907270939\)
\(L(3)\) 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 \( 1 + (3.32 + 8.36i)T \)
good5 \( 1 + 25.4iT - 625T^{2} \)
7 \( 1 - 36.1T + 2.40e3T^{2} \)
11 \( 1 + 188. iT - 1.46e4T^{2} \)
13 \( 1 - 223.T + 2.85e4T^{2} \)
17 \( 1 + 271. iT - 8.35e4T^{2} \)
19 \( 1 - 255.T + 1.30e5T^{2} \)
23 \( 1 + 308. iT - 2.79e5T^{2} \)
29 \( 1 + 1.04e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.13e3T + 9.23e5T^{2} \)
37 \( 1 + 1.42e3T + 1.87e6T^{2} \)
41 \( 1 - 1.58e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.58e3T + 3.41e6T^{2} \)
47 \( 1 + 430. iT - 4.87e6T^{2} \)
53 \( 1 - 316. iT - 7.89e6T^{2} \)
59 \( 1 - 5.79e3iT - 1.21e7T^{2} \)
61 \( 1 + 7.29e3T + 1.38e7T^{2} \)
67 \( 1 - 4.83e3T + 2.01e7T^{2} \)
71 \( 1 + 733. iT - 2.54e7T^{2} \)
73 \( 1 + 7.14e3T + 2.83e7T^{2} \)
79 \( 1 + 7.04e3T + 3.89e7T^{2} \)
83 \( 1 + 4.73e3iT - 4.74e7T^{2} \)
89 \( 1 + 581. iT - 6.27e7T^{2} \)
97 \( 1 - 7.16e3T + 8.85e7T^{2} \)
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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

−10.60675399393238868990839345191, −8.989390270017199672796314298195, −8.426593460803667942397785574914, −7.68841384542123795892065750766, −6.29682347628740886085171542822, −5.58785280477618941884314785563, −4.55040478478363472171100277754, −2.92597289188744347151432495669, −1.26872842880251755164345769072, −0.66002124624473258429684388643, 1.57147533259244726649121180670, 3.16823082939302566132668876734, 4.17441176073676109968756427530, 5.21308750940394651339428474088, 6.31277909884457781958194864660, 7.27114498941553225367947523440, 8.439674685300174240142073296656, 9.467347199630069566948943928062, 10.45913870585579658156244520274, 10.86136362586031920130643449645

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