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Properties

Label 2-384-48.35-c1-0-7
Degree $2$
Conductor $384$
Sign $0.517 + 0.855i$
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  + (−0.814 + 1.52i)3-s + (−2.08 − 2.08i)5-s + 1.14·7-s + (−1.67 − 2.48i)9-s + (1.67 − 1.67i)11-s + (−0.146 − 0.146i)13-s + (4.88 − 1.48i)15-s − 5.59i·17-s + (1.48 − 1.48i)19-s + (−0.933 + 1.75i)21-s − 3.34i·23-s + 3.68i·25-s + (5.16 − 0.533i)27-s + (3.51 − 3.51i)29-s + 5.83i·31-s + ⋯
L(s)  = 1  + (−0.470 + 0.882i)3-s + (−0.931 − 0.931i)5-s + 0.433·7-s + (−0.558 − 0.829i)9-s + (0.504 − 0.504i)11-s + (−0.0405 − 0.0405i)13-s + (1.26 − 0.384i)15-s − 1.35i·17-s + (0.341 − 0.341i)19-s + (−0.203 + 0.382i)21-s − 0.698i·23-s + 0.737i·25-s + (0.994 − 0.102i)27-s + (0.652 − 0.652i)29-s + 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.517 + 0.855i$
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: \((2,\ 384,\ (\ :1/2),\ 0.517 + 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762015 - 0.429936i\)
\(L(\frac12)\) \(\approx\) \(0.762015 - 0.429936i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.814 - 1.52i)T \)
good5 \( 1 + (2.08 + 2.08i)T + 5iT^{2} \)
7 \( 1 - 1.14T + 7T^{2} \)
11 \( 1 + (-1.67 + 1.67i)T - 11iT^{2} \)
13 \( 1 + (0.146 + 0.146i)T + 13iT^{2} \)
17 \( 1 + 5.59iT - 17T^{2} \)
19 \( 1 + (-1.48 + 1.48i)T - 19iT^{2} \)
23 \( 1 + 3.34iT - 23T^{2} \)
29 \( 1 + (-3.51 + 3.51i)T - 29iT^{2} \)
31 \( 1 - 5.83iT - 31T^{2} \)
37 \( 1 + (-4.83 + 4.83i)T - 37iT^{2} \)
41 \( 1 - 0.610T + 41T^{2} \)
43 \( 1 + (1.48 + 1.48i)T + 43iT^{2} \)
47 \( 1 + 6.41T + 47T^{2} \)
53 \( 1 + (0.164 + 0.164i)T + 53iT^{2} \)
59 \( 1 + (9.05 - 9.05i)T - 59iT^{2} \)
61 \( 1 + (4.53 + 4.53i)T + 61iT^{2} \)
67 \( 1 + (0.635 - 0.635i)T - 67iT^{2} \)
71 \( 1 - 6.90iT - 71T^{2} \)
73 \( 1 + 7.07iT - 73T^{2} \)
79 \( 1 + 9.83iT - 79T^{2} \)
83 \( 1 + (8.09 + 8.09i)T + 83iT^{2} \)
89 \( 1 - 0.490T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−11.46252719730160216974436498117, −10.34461870683309301118825032941, −9.221119909837008164020188018014, −8.644808832035634471222063845060, −7.58023165175903758162755122775, −6.25018760465023040238501292105, −4.95036023547958501342522291195, −4.48186109213216274342471452387, −3.20858165927500070271343252674, −0.65839663388712949252846630315, 1.65633286106228768207665421210, 3.26538733539309828222813562760, 4.55545798750987871590747854650, 5.98705880992549098969454616894, 6.82801183306922758156904265784, 7.70031953802668557621563274598, 8.299635113412023135157249344483, 9.838730779832661136862287780752, 10.95143034745703326690226363566, 11.44714366858023715278687380578

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