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Properties

Label 2-3328-13.2-c0-0-1
Degree $2$
Conductor $3328$
Sign $0.477 + 0.878i$
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  + (1.36 − 1.36i)5-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)13-s + (−0.866 − 0.5i)17-s − 2.73i·25-s + (0.5 + 0.866i)29-s + (−0.133 + 0.5i)37-s + (1.86 + 0.5i)41-s + (−0.499 − 1.86i)45-s + (−0.866 + 0.5i)49-s − 1.73·53-s + (−0.866 + 1.5i)61-s + (1.86 + 0.499i)65-s + (−1.36 − 1.36i)73-s + (−0.499 − 0.866i)81-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)5-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)13-s + (−0.866 − 0.5i)17-s − 2.73i·25-s + (0.5 + 0.866i)29-s + (−0.133 + 0.5i)37-s + (1.86 + 0.5i)41-s + (−0.499 − 1.86i)45-s + (−0.866 + 0.5i)49-s − 1.73·53-s + (−0.866 + 1.5i)61-s + (1.86 + 0.499i)65-s + (−1.36 − 1.36i)73-s + (−0.499 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.477 + 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.689687584\)
\(L(\frac12)\) \(\approx\) \(1.689687584\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
7 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + 1.73T + T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.068693128855887263304473704576, −8.158282535473652659874351432649, −7.00209298046707613382970475168, −6.29444915953773273099629052913, −5.79511055337411486745066181901, −4.61131212756556426212832585643, −4.44864083082921112652342133087, −2.99900469812904621976466513271, −1.81661158339523664562968314589, −1.11730136119031599691881538203, 1.66692955712819252594880612619, 2.40739622598879116668452152325, 3.18361625523954159725513623631, 4.30131547346905608110863132873, 5.33043145714906196876032397138, 6.05950919044436055927109039937, 6.54405698465401242224135561379, 7.41448260262178277553305254932, 8.051371014255892582543224590096, 9.073686226643975707758912240454

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