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Properties

Label 2-3264-408.101-c0-0-10
Degree $2$
Conductor $3264$
Sign $-0.965 + 0.258i$
Analytic cond. $1.62894$
Root an. cond. $1.27630$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s − 1.73i·5-s − 9-s − i·11-s − 1.73i·13-s − 1.73·15-s + 17-s + i·19-s + 1.73·23-s − 1.99·25-s + i·27-s − 33-s − 1.73·39-s − 41-s + i·43-s + ⋯
L(s)  = 1  − i·3-s − 1.73i·5-s − 9-s − i·11-s − 1.73i·13-s − 1.73·15-s + 17-s + i·19-s + 1.73·23-s − 1.99·25-s + i·27-s − 33-s − 1.73·39-s − 41-s + i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3264\)    =    \(2^{6} \cdot 3 \cdot 17\)
Sign: $-0.965 + 0.258i$
Analytic conductor: \(1.62894\)
Root analytic conductor: \(1.27630\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3264} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3264,\ (\ :0),\ -0.965 + 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.209486690\)
\(L(\frac12)\) \(\approx\) \(1.209486690\)
\(L(1)\) 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 \( 1 + iT \)
17 \( 1 - T \)
good5 \( 1 + 1.73iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + 1.73iT - T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 - 1.73T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 2iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.339427893449184186913383686140, −7.996209660083021062957586083404, −7.17829844542105445598273047106, −5.91416963227509108683541007991, −5.57101828514612178029399779433, −4.95565659862843671993096136281, −3.60021145163613694654284618606, −2.83857954611658931078385087679, −1.31852263236186168671539282407, −0.816559878042414031355438124224, 2.03545759875358866526578233590, 2.90475845679896136603735050647, 3.61665002572117307216603972560, 4.49286412530292189580516121540, 5.23734264122509667623065715603, 6.32458731494835619276285996614, 6.99090093487223198203169423796, 7.36531894427687427422716380226, 8.627069889418799967986850455421, 9.388136409169402434347896764819

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