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Properties

Label 2-1920-120.83-c0-0-3
Degree $2$
Conductor $1920$
Sign $0.229 + 0.973i$
Analytic cond. $0.958204$
Root an. cond. $0.978879$
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  + (0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s + (−1 − i)7-s − 1.00i·9-s − 1.41i·11-s + 1.00·15-s − 1.41·21-s + 1.00i·25-s + (−0.707 − 0.707i)27-s − 1.41i·29-s + 2i·31-s + (−1.00 − 1.00i)33-s − 1.41i·35-s + (0.707 − 0.707i)45-s + i·49-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s + (−1 − i)7-s − 1.00i·9-s − 1.41i·11-s + 1.00·15-s − 1.41·21-s + 1.00i·25-s + (−0.707 − 0.707i)27-s − 1.41i·29-s + 2i·31-s + (−1.00 − 1.00i)33-s − 1.41i·35-s + (0.707 − 0.707i)45-s + i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.430779601\)
\(L(\frac12)\) \(\approx\) \(1.430779601\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (1 + i)T + iT^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 - 2iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−9.235013473790616003098099410677, −8.419853316878536067506712252319, −7.59866948321653186934864673513, −6.66959394052918352607921649706, −6.45731506835580998925713644539, −5.45923872534482684778811878080, −3.81210466654432587232686181499, −3.28023380292448140972973244351, −2.41901638947738006320983494939, −0.989495845916866369927053587124, 1.94802535147374104621354704193, 2.60758888658230569363975613671, 3.75813702065245942493599098377, 4.72903011280421895005671350802, 5.40806397964301560766050498366, 6.27174018358872817886945580480, 7.27325772261590148442805856320, 8.259627556676278746324442074741, 9.016085130355660015946740466055, 9.591924143184101348905935909882

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