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Properties

Label 2-384-48.35-c1-0-4
Degree $2$
Conductor $384$
Sign $0.995 + 0.0990i$
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  + (−0.966 − 1.43i)3-s + (1.57 + 1.57i)5-s + 2.24·7-s + (−1.13 + 2.77i)9-s + (−1.13 + 1.13i)11-s + (3.24 + 3.24i)13-s + (0.739 − 3.77i)15-s − 1.66i·17-s + (3.77 − 3.77i)19-s + (−2.17 − 3.23i)21-s + 2.26i·23-s − 0.0586i·25-s + (5.08 − 1.05i)27-s + (3.23 − 3.23i)29-s + 1.30i·31-s + ⋯
L(s)  = 1  + (−0.558 − 0.829i)3-s + (0.702 + 0.702i)5-s + 0.850·7-s + (−0.377 + 0.926i)9-s + (−0.341 + 0.341i)11-s + (0.901 + 0.901i)13-s + (0.191 − 0.975i)15-s − 0.403i·17-s + (0.866 − 0.866i)19-s + (−0.474 − 0.705i)21-s + 0.471i·23-s − 0.0117i·25-s + (0.978 − 0.203i)27-s + (0.600 − 0.600i)29-s + 0.234i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.995 + 0.0990i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.995 + 0.0990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39532 - 0.0692879i\)
\(L(\frac12)\) \(\approx\) \(1.39532 - 0.0692879i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.966 + 1.43i)T \)
good5 \( 1 + (-1.57 - 1.57i)T + 5iT^{2} \)
7 \( 1 - 2.24T + 7T^{2} \)
11 \( 1 + (1.13 - 1.13i)T - 11iT^{2} \)
13 \( 1 + (-3.24 - 3.24i)T + 13iT^{2} \)
17 \( 1 + 1.66iT - 17T^{2} \)
19 \( 1 + (-3.77 + 3.77i)T - 19iT^{2} \)
23 \( 1 - 2.26iT - 23T^{2} \)
29 \( 1 + (-3.23 + 3.23i)T - 29iT^{2} \)
31 \( 1 - 1.30iT - 31T^{2} \)
37 \( 1 + (2.30 - 2.30i)T - 37iT^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + (3.77 + 3.77i)T + 43iT^{2} \)
47 \( 1 + 3.74T + 47T^{2} \)
53 \( 1 + (0.972 + 0.972i)T + 53iT^{2} \)
59 \( 1 + (3.88 - 3.88i)T - 59iT^{2} \)
61 \( 1 + (4.19 + 4.19i)T + 61iT^{2} \)
67 \( 1 + (8.02 - 8.02i)T - 67iT^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 + 6.38iT - 73T^{2} \)
79 \( 1 - 2.69iT - 79T^{2} \)
83 \( 1 + (2.61 + 2.61i)T + 83iT^{2} \)
89 \( 1 + 7.35T + 89T^{2} \)
97 \( 1 + 5.67T + 97T^{2} \)
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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

−11.35645031904019563091086668130, −10.67340114903256670067633386067, −9.580175268802677294048187199960, −8.405113261952221436906444476021, −7.37674569756438379216376073314, −6.61779114003020429169071474229, −5.67119133181959981570248255369, −4.62662671339790683852648582256, −2.69456275476230197630858469209, −1.50364311978344121610032969637, 1.27117601779801645650005727297, 3.31327204217960823529705412766, 4.64711265552733013633045759295, 5.49320910113251554518116332095, 6.12962022223807663568513509062, 7.87483105930917630780980365266, 8.687463909962237954468343028311, 9.604477562894527900070905024132, 10.54966992980409660957455796918, 11.11322618798679536756595848361

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