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Properties

Label 2-3328-52.51-c0-0-6
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

\(L(\frac{1}{2})\) \(\approx\) \(1.462936171\)
\(L(\frac12)\) \(\approx\) \(1.462936171\)
\(L(1)\) 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

−8.477680431705085393959180377892, −8.050892922847560040132224937998, −7.18410445313505903383905046371, −6.62646188260096688472063870144, −6.23591269333276217380344902089, −4.83271461194705161770510980675, −4.31600904257227200002021443325, −2.98490040533433878531851990150, −2.15525573771881859485763451079, −1.34589759921834001630350888646, 1.02774226071832886268887575312, 2.25140568163379909471755346637, 3.49074561355108504430324564804, 4.43275764455333308459180915377, 4.84265861513939531927462503886, 5.42212603440099037756837677185, 6.47943368655451322529920435378, 7.53151258967871257381789678345, 8.309833821496062188442573102793, 8.740799235582977197137723489628

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