Properties

Label 2-384-384.11-c1-0-15
Degree $2$
Conductor $384$
Sign $0.690 - 0.723i$
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  + (−1.30 − 0.555i)2-s + (−1.67 + 0.423i)3-s + (1.38 + 1.44i)4-s + (3.35 + 1.79i)5-s + (2.41 + 0.382i)6-s + (3.20 + 0.637i)7-s + (−0.995 − 2.64i)8-s + (2.64 − 1.42i)9-s + (−3.36 − 4.19i)10-s + (−2.82 + 2.31i)11-s + (−2.93 − 1.84i)12-s + (−3.67 + 1.96i)13-s + (−3.81 − 2.60i)14-s + (−6.39 − 1.59i)15-s + (−0.175 + 3.99i)16-s + (4.87 − 2.01i)17-s + ⋯
L(s)  = 1  + (−0.919 − 0.392i)2-s + (−0.969 + 0.244i)3-s + (0.691 + 0.722i)4-s + (1.49 + 0.801i)5-s + (0.987 + 0.156i)6-s + (1.21 + 0.240i)7-s + (−0.352 − 0.935i)8-s + (0.880 − 0.473i)9-s + (−1.06 − 1.32i)10-s + (−0.850 + 0.698i)11-s + (−0.846 − 0.531i)12-s + (−1.01 + 0.544i)13-s + (−1.01 − 0.697i)14-s + (−1.65 − 0.411i)15-s + (−0.0438 + 0.999i)16-s + (1.18 − 0.489i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.690 - 0.723i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.690 - 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.845115 + 0.361382i\)
\(L(\frac12)\) \(\approx\) \(0.845115 + 0.361382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 + 0.555i)T \)
3 \( 1 + (1.67 - 0.423i)T \)
good5 \( 1 + (-3.35 - 1.79i)T + (2.77 + 4.15i)T^{2} \)
7 \( 1 + (-3.20 - 0.637i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (2.82 - 2.31i)T + (2.14 - 10.7i)T^{2} \)
13 \( 1 + (3.67 - 1.96i)T + (7.22 - 10.8i)T^{2} \)
17 \( 1 + (-4.87 + 2.01i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-1.12 + 3.70i)T + (-15.7 - 10.5i)T^{2} \)
23 \( 1 + (0.487 + 0.326i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (-4.48 - 3.68i)T + (5.65 + 28.4i)T^{2} \)
31 \( 1 + (4.98 - 4.98i)T - 31iT^{2} \)
37 \( 1 + (2.84 + 9.37i)T + (-30.7 + 20.5i)T^{2} \)
41 \( 1 + (-6.15 - 4.11i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (-0.0174 + 0.177i)T + (-42.1 - 8.38i)T^{2} \)
47 \( 1 + (0.894 - 0.370i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (4.98 - 4.08i)T + (10.3 - 51.9i)T^{2} \)
59 \( 1 + (-0.788 - 0.421i)T + (32.7 + 49.0i)T^{2} \)
61 \( 1 + (-1.36 - 13.8i)T + (-59.8 + 11.9i)T^{2} \)
67 \( 1 + (-5.73 + 0.565i)T + (65.7 - 13.0i)T^{2} \)
71 \( 1 + (-3.74 - 0.745i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (1.19 + 6.01i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (-0.130 + 0.314i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-1.53 + 5.07i)T + (-69.0 - 46.1i)T^{2} \)
89 \( 1 + (2.05 + 3.07i)T + (-34.0 + 82.2i)T^{2} \)
97 \( 1 + (2.84 + 2.84i)T + 97iT^{2} \)
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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

−11.12725506318468339309655329605, −10.50449734887894819748695849145, −9.841603977704052062944515929918, −9.122540239198260065735647941735, −7.49762697421205146329773563387, −6.96577069579948395952919938023, −5.64568198236059257609486066662, −4.84262216791950037303338164701, −2.70252288284928672257506811729, −1.63779962818745576863601492028, 1.02096498775639029870354023161, 2.09755472281896885934566476638, 5.09532990904885623585203167423, 5.41376928012505586487992027247, 6.28471350587836255359723957673, 7.77881897333296430098676035430, 8.163614732772436636099863033244, 9.681948327109631428306947848830, 10.15781170725806025059032285374, 10.93719120553437517632053867343

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