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Properties

Label 2-3328-52.51-c0-0-12
Degree $2$
Conductor $3328$
Sign $-1$
Analytic cond. $1.66088$
Root an. cond. $1.28875$
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 − i·5-s − 7-s − i·13-s − 15-s − 17-s + i·21-s − i·27-s − 2·31-s + i·35-s − i·37-s − 39-s + i·43-s + 47-s + i·51-s + ⋯
L(s)  = 1  − i·3-s − i·5-s − 7-s − i·13-s − 15-s − 17-s + i·21-s − i·27-s − 2·31-s + i·35-s − i·37-s − 39-s + i·43-s + 47-s + i·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (3327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7906141331\)
\(L(\frac12)\) \(\approx\) \(0.7906141331\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + iT \)
good3 \( 1 + iT - T^{2} \)
5 \( 1 + iT - T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 2T + T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T + T^{2} \)
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

−8.456659332208440573307647116278, −7.55995553856146921579089603365, −7.06980072597460432043684858967, −6.19298973486901116145811799948, −5.57766944819752483179688164989, −4.61457999541250677639629245388, −3.69267487646518361871695322462, −2.63102501446484539378531365409, −1.60575815015754326078566069670, −0.45468294299043555927308031050, 1.96319392363277027465208035418, 3.01423974435802434818444476930, 3.73291840204906452544089757887, 4.37153898682535228787541908803, 5.34318332488671053338281874127, 6.34227980712601482043790375220, 6.87962826967830132674995307973, 7.44208257964145552273456310161, 8.857798240330498516345391000126, 9.163644565725762270798183418088

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