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Properties

Label 2-3328-104.75-c0-0-0
Degree $2$
Conductor $3328$
Sign $0.271 - 0.962i$
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  − 1.73·5-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)17-s + 1.99·25-s + (0.866 − 0.5i)29-s + (0.866 + 1.5i)37-s + (−1.5 + 0.866i)41-s + (−0.866 + 1.49i)45-s + (0.5 + 0.866i)49-s + i·53-s + (0.866 + 0.5i)61-s + (1.49 − 0.866i)65-s + 1.73i·73-s + (−0.499 − 0.866i)81-s + ⋯
L(s)  = 1  − 1.73·5-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)17-s + 1.99·25-s + (0.866 − 0.5i)29-s + (0.866 + 1.5i)37-s + (−1.5 + 0.866i)41-s + (−0.866 + 1.49i)45-s + (0.5 + 0.866i)49-s + i·53-s + (0.866 + 0.5i)61-s + (1.49 − 0.866i)65-s + 1.73i·73-s + (−0.499 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.271 - 0.962i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.271 - 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6418069190\)
\(L(\frac12)\) \(\approx\) \(0.6418069190\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + 1.73T + T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−8.737778393875349211763520342750, −8.238954138908719196074383234938, −7.46752865760807208924853084955, −6.83723285174300044084969009779, −6.20010486439529494742274123325, −4.77966099739463954391516614087, −4.31701411532477413659281682714, −3.61460755464602750014414060052, −2.68361516535691100717612486492, −1.13546622096419668590799659653, 0.44609530053927863753975185683, 2.17343518217213952893652564310, 3.16325958183594263869775798770, 4.03615759247945831920055140807, 4.78419085713673820792893270654, 5.31002565227943186156918971770, 6.80497548799002166698439251939, 7.23663566486455935610289317753, 7.87658509555121710982444043386, 8.431500447140368024274152741020

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