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Properties

Label 2-384-128.29-c1-0-9
Degree $2$
Conductor $384$
Sign $-0.669 - 0.742i$
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  + (−0.957 + 1.04i)2-s + (−0.0980 + 0.995i)3-s + (−0.166 − 1.99i)4-s + (0.269 − 0.143i)5-s + (−0.941 − 1.05i)6-s + (0.809 + 4.07i)7-s + (2.23 + 1.73i)8-s + (−0.980 − 0.195i)9-s + (−0.108 + 0.418i)10-s + (3.25 − 3.96i)11-s + (1.99 + 0.0294i)12-s + (−0.832 + 1.55i)13-s + (−5.01 − 3.05i)14-s + (0.116 + 0.282i)15-s + (−3.94 + 0.664i)16-s + (−1.05 + 2.54i)17-s + ⋯
L(s)  = 1  + (−0.676 + 0.735i)2-s + (−0.0565 + 0.574i)3-s + (−0.0833 − 0.996i)4-s + (0.120 − 0.0643i)5-s + (−0.384 − 0.430i)6-s + (0.306 + 1.53i)7-s + (0.789 + 0.613i)8-s + (−0.326 − 0.0650i)9-s + (−0.0341 + 0.132i)10-s + (0.981 − 1.19i)11-s + (0.577 + 0.00848i)12-s + (−0.231 + 0.432i)13-s + (−1.33 − 0.816i)14-s + (0.0301 + 0.0728i)15-s + (−0.986 + 0.166i)16-s + (−0.255 + 0.616i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.669 - 0.742i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.669 - 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.378557 + 0.851284i\)
\(L(\frac12)\) \(\approx\) \(0.378557 + 0.851284i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.957 - 1.04i)T \)
3 \( 1 + (0.0980 - 0.995i)T \)
good5 \( 1 + (-0.269 + 0.143i)T + (2.77 - 4.15i)T^{2} \)
7 \( 1 + (-0.809 - 4.07i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-3.25 + 3.96i)T + (-2.14 - 10.7i)T^{2} \)
13 \( 1 + (0.832 - 1.55i)T + (-7.22 - 10.8i)T^{2} \)
17 \( 1 + (1.05 - 2.54i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-1.54 - 5.09i)T + (-15.7 + 10.5i)T^{2} \)
23 \( 1 + (-2.66 + 1.78i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (4.96 - 4.07i)T + (5.65 - 28.4i)T^{2} \)
31 \( 1 + (3.88 - 3.88i)T - 31iT^{2} \)
37 \( 1 + (-0.663 - 0.201i)T + (30.7 + 20.5i)T^{2} \)
41 \( 1 + (4.02 + 6.01i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.733 + 7.44i)T + (-42.1 + 8.38i)T^{2} \)
47 \( 1 + (2.51 + 1.04i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (-10.1 - 8.33i)T + (10.3 + 51.9i)T^{2} \)
59 \( 1 + (-0.305 - 0.571i)T + (-32.7 + 49.0i)T^{2} \)
61 \( 1 + (-7.82 - 0.770i)T + (59.8 + 11.9i)T^{2} \)
67 \( 1 + (-13.0 - 1.28i)T + (65.7 + 13.0i)T^{2} \)
71 \( 1 + (-5.38 + 1.07i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-1.26 + 6.33i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-10.1 + 4.18i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (6.59 - 2.00i)T + (69.0 - 46.1i)T^{2} \)
89 \( 1 + (5.61 + 3.75i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (4.68 - 4.68i)T - 97iT^{2} \)
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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

−11.45732176854588750381738633444, −10.67978773075887577997684147844, −9.471524453662464117985474309943, −8.879935718627490439978440188462, −8.355434218078538611595061217526, −6.90361280803606193483247291011, −5.75030407685250670935384717958, −5.37359042649533159387907020224, −3.70333329431624017849492486299, −1.82471034585301627456270682974, 0.827005155328200141973416701329, 2.19475557380381418543694954900, 3.77531211327111480560095532693, 4.77256893289973628640628177991, 6.81845089329005897098258078961, 7.25794197398751809226209157017, 8.136637204285671456970154993057, 9.522651428018014514389376873501, 9.926833216412639967767374933277, 11.25011092725161434108807264261

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