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Properties

Label 2-1920-120.83-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.131104781\)
\(L(\frac12)\) \(\approx\) \(1.131104781\)
\(L(1)\) 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

−9.709195170997913339828781332842, −9.142806128109748531150309281721, −8.032501818761722075677661211892, −7.18537040945745277514709759233, −6.18964141348491523536799112005, −5.65848502440513490006599400531, −4.81327369953146661114278623556, −4.08819356977308668528192630931, −2.63462546601600168283344915401, −1.83517595269301643485706422095, 1.00002870577808294262421844244, 1.68755528273253919407374476225, 3.20996261369578796104399443215, 4.58600373420859137414224727777, 5.16524194696221012679865300948, 5.93228445161570808009855990297, 6.76708086808752932953988665360, 7.57460504842845185962095510032, 8.414842970110386394010797131375, 8.890507703794738213465800479708

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