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Properties

Label 2-3328-104.37-c0-0-0
Degree $2$
Conductor $3328$
Sign $-0.955 + 0.295i$
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 + (−1.5 + 0.866i)29-s + (−0.133 + 0.5i)37-s + (0.133 − 0.5i)41-s + (−0.499 − 1.86i)45-s + (0.866 − 0.5i)49-s − i·53-s + (−0.866 − 0.5i)61-s + (−1.86 − 0.499i)65-s + (−0.366 + 0.366i)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 + (−1.5 + 0.866i)29-s + (−0.133 + 0.5i)37-s + (0.133 − 0.5i)41-s + (−0.499 − 1.86i)45-s + (0.866 − 0.5i)49-s − i·53-s + (−0.866 − 0.5i)61-s + (−1.86 − 0.499i)65-s + (−0.366 + 0.366i)73-s + (−0.499 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $-0.955 + 0.295i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (2689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ -0.955 + 0.295i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3341292833\)
\(L(\frac12)\) \(\approx\) \(0.3341292833\)
\(L(1)\) 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 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.013200620594773791345739815023, −8.397406401181680933437308207660, −7.60863058403809507127022986418, −7.05541077041626817501298720596, −6.49073243136426039897523368682, −5.42406597786753738341249211891, −4.41448454533827495339108732711, −3.73456924262829059320673938859, −2.90968746262043890025891281215, −2.02349332797836632762989314853, 0.20494140139660040004872974889, 1.35768153120361413018732978728, 2.94473824988243938028724742329, 3.95426334084597169038563067688, 4.24946295514276507345539100120, 5.41497228566672551810705747596, 5.94981668800274751717051396923, 7.07547891958501670784966977019, 7.88022488675572606176309312258, 8.366538202623736089378268335280

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