Properties

Label 2-384-3.2-c2-0-7
Degree $2$
Conductor $384$
Sign $0.850 - 0.526i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.57 − 2.55i)3-s − 1.31i·5-s − 10.2·7-s + (−4.01 + 8.05i)9-s + 16.6i·11-s + 18.7·13-s + (−3.35 + 2.07i)15-s + 4.38i·17-s + 11.5·19-s + (16.1 + 26.1i)21-s − 16.7i·23-s + 23.2·25-s + (26.8 − 2.46i)27-s − 12.5i·29-s − 20.3·31-s + ⋯
L(s)  = 1  + (−0.526 − 0.850i)3-s − 0.263i·5-s − 1.46·7-s + (−0.446 + 0.894i)9-s + 1.51i·11-s + 1.44·13-s + (−0.223 + 0.138i)15-s + 0.257i·17-s + 0.608·19-s + (0.769 + 1.24i)21-s − 0.728i·23-s + 0.930·25-s + (0.995 − 0.0912i)27-s − 0.432i·29-s − 0.655·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.850 - 0.526i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.850 - 0.526i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.951897 + 0.270665i\)
\(L(\frac12)\) \(\approx\) \(0.951897 + 0.270665i\)
\(L(2)\) 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 \( 1 + (1.57 + 2.55i)T \)
good5 \( 1 + 1.31iT - 25T^{2} \)
7 \( 1 + 10.2T + 49T^{2} \)
11 \( 1 - 16.6iT - 121T^{2} \)
13 \( 1 - 18.7T + 169T^{2} \)
17 \( 1 - 4.38iT - 289T^{2} \)
19 \( 1 - 11.5T + 361T^{2} \)
23 \( 1 + 16.7iT - 529T^{2} \)
29 \( 1 + 12.5iT - 841T^{2} \)
31 \( 1 + 20.3T + 961T^{2} \)
37 \( 1 + 18.5T + 1.36e3T^{2} \)
41 \( 1 - 78.6iT - 1.68e3T^{2} \)
43 \( 1 - 36.4T + 1.84e3T^{2} \)
47 \( 1 - 19.9iT - 2.20e3T^{2} \)
53 \( 1 - 81.3iT - 2.80e3T^{2} \)
59 \( 1 - 29.9iT - 3.48e3T^{2} \)
61 \( 1 - 72.0T + 3.72e3T^{2} \)
67 \( 1 - 56.3T + 4.48e3T^{2} \)
71 \( 1 - 136. iT - 5.04e3T^{2} \)
73 \( 1 + 80.8T + 5.32e3T^{2} \)
79 \( 1 + 86.0T + 6.24e3T^{2} \)
83 \( 1 - 80.4iT - 6.88e3T^{2} \)
89 \( 1 - 131. iT - 7.92e3T^{2} \)
97 \( 1 + 20.4T + 9.40e3T^{2} \)
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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

−11.28414596709055785211403997896, −10.31876798200679797823084920694, −9.422763381927462918408093508814, −8.376026040463315345563030443672, −7.18301646467573892167962444446, −6.52800684440847176081238032070, −5.64423585903623719660018869940, −4.26346289360657038071459306105, −2.77373104944893571037160484838, −1.18776845487793353497849622326, 0.55036363486887212782945184508, 3.32437891016868888822833385207, 3.58681651553584042806156545902, 5.42655032358615403474507116527, 6.08705978256123349581151783304, 6.97081325847389003210229237950, 8.665369355269045341633405314249, 9.200510555636234671281163620030, 10.27896002279338177162270061582, 10.94465330432287389318633637142

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