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Properties

Label 2-3328-104.3-c0-0-1
Degree $2$
Conductor $3328$
Sign $0.869 + 0.494i$
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  + (−0.5 − 0.866i)3-s + (0.866 + 0.5i)7-s + (0.5 + 0.866i)11-s − i·13-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 0.999i·21-s + (0.866 − 0.5i)23-s + 25-s − 27-s + (0.866 − 0.5i)29-s + (0.499 − 0.866i)33-s + (0.866 − 0.5i)37-s + (−0.866 + 0.5i)39-s + (0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.866 + 0.5i)7-s + (0.5 + 0.866i)11-s − i·13-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 0.999i·21-s + (0.866 − 0.5i)23-s + 25-s − 27-s + (0.866 − 0.5i)29-s + (0.499 − 0.866i)33-s + (0.866 − 0.5i)37-s + (−0.866 + 0.5i)39-s + (0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.869 + 0.494i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.869 + 0.494i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.250571734\)
\(L(\frac12)\) \(\approx\) \(1.250571734\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.548789363272009092083044346728, −7.950405011517662337077733027513, −7.29624914958772227858701424376, −6.28799208113674810285621194433, −6.08482032941977788700925707835, −4.87671908976711168355209788077, −4.33743669548495151776935855339, −3.01053603574035786523030628492, −1.93126743368904619955514588949, −1.12827592633435009775254966264, 1.05625640460869781486493385536, 2.38014076217482801614434195795, 3.57020999915190925212135949132, 4.48468381298464981254754130596, 4.83137606725482634088279880017, 5.64156120340410846514707989308, 6.83025357173996165585776276487, 7.10856046967053938299017052053, 8.343708900037541706845941557258, 8.923939827776000523788138207180

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