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Properties

Label 2-384-128.13-c1-0-26
Degree $2$
Conductor $384$
Sign $-0.327 + 0.944i$
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  + (−0.156 + 1.40i)2-s + (−0.290 − 0.956i)3-s + (−1.95 − 0.439i)4-s + (−0.103 − 1.05i)5-s + (1.39 − 0.258i)6-s + (−0.0783 + 0.117i)7-s + (0.922 − 2.67i)8-s + (−0.831 + 0.555i)9-s + (1.49 + 0.0188i)10-s + (−2.29 + 1.22i)11-s + (0.146 + 1.99i)12-s + (−6.11 − 0.602i)13-s + (−0.152 − 0.128i)14-s + (−0.979 + 0.405i)15-s + (3.61 + 1.71i)16-s + (−6.95 − 2.88i)17-s + ⋯
L(s)  = 1  + (−0.110 + 0.993i)2-s + (−0.167 − 0.552i)3-s + (−0.975 − 0.219i)4-s + (−0.0464 − 0.471i)5-s + (0.567 − 0.105i)6-s + (−0.0295 + 0.0442i)7-s + (0.326 − 0.945i)8-s + (−0.277 + 0.185i)9-s + (0.473 + 0.00594i)10-s + (−0.692 + 0.370i)11-s + (0.0421 + 0.575i)12-s + (−1.69 − 0.167i)13-s + (−0.0407 − 0.0343i)14-s + (−0.252 + 0.104i)15-s + (0.903 + 0.428i)16-s + (−1.68 − 0.698i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.327 + 0.944i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.327 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.215160 - 0.302174i\)
\(L(\frac12)\) \(\approx\) \(0.215160 - 0.302174i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.156 - 1.40i)T \)
3 \( 1 + (0.290 + 0.956i)T \)
good5 \( 1 + (0.103 + 1.05i)T + (-4.90 + 0.975i)T^{2} \)
7 \( 1 + (0.0783 - 0.117i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (2.29 - 1.22i)T + (6.11 - 9.14i)T^{2} \)
13 \( 1 + (6.11 + 0.602i)T + (12.7 + 2.53i)T^{2} \)
17 \( 1 + (6.95 + 2.88i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (-0.671 + 0.551i)T + (3.70 - 18.6i)T^{2} \)
23 \( 1 + (-1.85 + 9.34i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (3.35 - 6.27i)T + (-16.1 - 24.1i)T^{2} \)
31 \( 1 + (-3.67 + 3.67i)T - 31iT^{2} \)
37 \( 1 + (1.42 - 1.73i)T + (-7.21 - 36.2i)T^{2} \)
41 \( 1 + (6.97 + 1.38i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (0.779 - 2.56i)T + (-35.7 - 23.8i)T^{2} \)
47 \( 1 + (-0.674 + 1.62i)T + (-33.2 - 33.2i)T^{2} \)
53 \( 1 + (4.02 + 7.52i)T + (-29.4 + 44.0i)T^{2} \)
59 \( 1 + (10.7 - 1.05i)T + (57.8 - 11.5i)T^{2} \)
61 \( 1 + (-6.46 + 1.96i)T + (50.7 - 33.8i)T^{2} \)
67 \( 1 + (-10.0 + 3.04i)T + (55.7 - 37.2i)T^{2} \)
71 \( 1 + (-9.98 - 6.67i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (-1.28 - 1.91i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (5.10 + 12.3i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-6.80 - 8.28i)T + (-16.1 + 81.4i)T^{2} \)
89 \( 1 + (1.23 + 6.19i)T + (-82.2 + 34.0i)T^{2} \)
97 \( 1 + (12.5 - 12.5i)T - 97iT^{2} \)
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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

−10.97926391060686095192595830550, −9.933872899578054272100471205550, −8.976767088436196866598580431127, −8.173586352228607766679753269623, −7.12947744249716208506733368892, −6.60296025139279815722590678646, −5.02882534861461348805121347379, −4.73395250108233856839511436731, −2.51062160917405320821724487164, −0.24654022820828621041164436844, 2.23908745210715374637408727604, 3.36094652232262510232842919777, 4.57215353235666521476316811526, 5.45284071698453995231214980253, 7.00991705867611306788245509658, 8.128712627882594153180580525716, 9.227738003310690629774109058932, 9.926283769427070472451382128349, 10.77825637514713950540200686965, 11.38425998795289054765102374663

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