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Properties

Label 2-1920-240.149-c0-0-3
Degree $2$
Conductor $1920$
Sign $-0.382 + 0.923i$
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 − 1.00i·9-s − 1.00·15-s − 1.41·17-s + (1 − i)19-s − 1.41i·23-s + 1.00i·25-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)45-s + 1.41·47-s − 49-s + (−1.00 + 1.00i)51-s − 1.41i·57-s + (−1 + i)61-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s − 1.00i·9-s − 1.00·15-s − 1.41·17-s + (1 − i)19-s − 1.41i·23-s + 1.00i·25-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)45-s + 1.41·47-s − 49-s + (−1.00 + 1.00i)51-s − 1.41i·57-s + (−1 + i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.152884520\)
\(L(\frac12)\) \(\approx\) \(1.152884520\)
\(L(1)\) 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 \( 1 + T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 + (-1 + i)T - iT^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (1 - i)T - iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.897388278363709874948411656893, −8.498316364981478303411875626758, −7.58309357102454823863004241707, −6.99616676341377960169103458985, −6.15482831036092311680169594509, −4.89964597872888503742949498674, −4.21035481134347670454651788897, −3.13451694512454482195420610889, −2.17186569299943386037840278972, −0.77884833184188450067356919233, 1.95778654842782115418464481243, 3.08555575098576558348423473765, 3.72322816787976880019680435729, 4.54147855639560133824732523319, 5.53044713783399615094760808943, 6.60406978827796684763351198882, 7.55273875562359617614173084054, 7.962387441856983550635079094090, 8.942714261574991466509451618012, 9.571572762585456116195893508332

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