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Properties

Label 2-1920-80.67-c1-0-41
Degree $2$
Conductor $1920$
Sign $-0.613 + 0.789i$
Analytic cond. $15.3312$
Root an. cond. $3.91551$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (0.609 − 2.15i)5-s + (0.566 + 0.566i)7-s − 9-s + (−3.64 − 3.64i)11-s + 2.74·13-s + (2.15 + 0.609i)15-s + (2.08 + 2.08i)17-s + (−5.79 − 5.79i)19-s + (−0.566 + 0.566i)21-s + (−4.28 + 4.28i)23-s + (−4.25 − 2.62i)25-s − i·27-s + (−2.63 + 2.63i)29-s − 8.10i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.272 − 0.962i)5-s + (0.214 + 0.214i)7-s − 0.333·9-s + (−1.09 − 1.09i)11-s + 0.760·13-s + (0.555 + 0.157i)15-s + (0.505 + 0.505i)17-s + (−1.32 − 1.32i)19-s + (−0.123 + 0.123i)21-s + (−0.892 + 0.892i)23-s + (−0.851 − 0.524i)25-s − 0.192i·27-s + (−0.489 + 0.489i)29-s − 1.45i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.613 + 0.789i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ -0.613 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8387817371\)
\(L(\frac12)\) \(\approx\) \(0.8387817371\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (-0.609 + 2.15i)T \)
good7 \( 1 + (-0.566 - 0.566i)T + 7iT^{2} \)
11 \( 1 + (3.64 + 3.64i)T + 11iT^{2} \)
13 \( 1 - 2.74T + 13T^{2} \)
17 \( 1 + (-2.08 - 2.08i)T + 17iT^{2} \)
19 \( 1 + (5.79 + 5.79i)T + 19iT^{2} \)
23 \( 1 + (4.28 - 4.28i)T - 23iT^{2} \)
29 \( 1 + (2.63 - 2.63i)T - 29iT^{2} \)
31 \( 1 + 8.10iT - 31T^{2} \)
37 \( 1 - 2.28T + 37T^{2} \)
41 \( 1 - 2.27iT - 41T^{2} \)
43 \( 1 + 3.06T + 43T^{2} \)
47 \( 1 + (1.80 - 1.80i)T - 47iT^{2} \)
53 \( 1 - 6.32iT - 53T^{2} \)
59 \( 1 + (5.56 - 5.56i)T - 59iT^{2} \)
61 \( 1 + (4.82 + 4.82i)T + 61iT^{2} \)
67 \( 1 + 3.34T + 67T^{2} \)
71 \( 1 + 2.81T + 71T^{2} \)
73 \( 1 + (10.7 + 10.7i)T + 73iT^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + 1.97iT - 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + (1.02 + 1.02i)T + 97iT^{2} \)
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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

−8.859091854464964047417047306015, −8.322538899131571765279122066978, −7.67372188599854003438478699520, −6.09678335470837403468399692591, −5.77751369409300002665221255429, −4.85391915188864612619736820027, −4.06389206077893432835517517233, −3.01423994271802970071453435321, −1.80940802886379468712955103043, −0.28224838032415902157166295126, 1.69596306063600972339445650089, 2.45381937734027330020887561432, 3.53850523536078556020986154154, 4.58672591681878957884047545747, 5.70095414795233902344182356209, 6.36352346536470650469562610290, 7.14795132398772159138052134682, 7.86373480795585482758131496988, 8.423296415393541172645764044157, 9.655713425768943489350631421365

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