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Properties

Label 2-1920-80.27-c1-0-37
Degree $2$
Conductor $1920$
Sign $0.937 + 0.348i$
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  + 3-s + (2.15 + 0.609i)5-s + (−0.566 − 0.566i)7-s + 9-s + (3.64 − 3.64i)11-s − 2.74i·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 + 27-s + (2.63 + 2.63i)29-s − 8.10i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.962 + 0.272i)5-s + (−0.214 − 0.214i)7-s + 0.333·9-s + (1.09 − 1.09i)11-s − 0.760i·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 + 0.192·27-s + (0.489 + 0.489i)29-s − 1.45i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.937 + 0.348i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.937 + 0.348i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.738710471\)
\(L(\frac12)\) \(\approx\) \(2.738710471\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + (-2.15 - 0.609i)T \)
good7 \( 1 + (0.566 + 0.566i)T + 7iT^{2} \)
11 \( 1 + (-3.64 + 3.64i)T - 11iT^{2} \)
13 \( 1 + 2.74iT - 13T^{2} \)
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 \( 1 - 2.28iT - 37T^{2} \)
41 \( 1 + 2.27iT - 41T^{2} \)
43 \( 1 + 3.06iT - 43T^{2} \)
47 \( 1 + (1.80 - 1.80i)T - 47iT^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 + (5.56 + 5.56i)T + 59iT^{2} \)
61 \( 1 + (4.82 - 4.82i)T - 61iT^{2} \)
67 \( 1 - 3.34iT - 67T^{2} \)
71 \( 1 - 2.81T + 71T^{2} \)
73 \( 1 + (-10.7 - 10.7i)T + 73iT^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + 1.97T + 83T^{2} \)
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

−9.071671638405057761532791792606, −8.490681486967995319015040976737, −7.75212165753208838337746306935, −6.40871990996597046166700143110, −6.31719265364501913531784685201, −5.22459052811244121994563755050, −3.94304474120578797039616313244, −3.28038950991628692764715404153, −2.21408646170953087560274327144, −1.06944998434389685897470673899, 1.36521505059875924075895079591, 2.21741027652600697184903881549, 3.22288506160862634998887369400, 4.50622214861858612013395938425, 4.96237969562876782091896895462, 6.37578972967972955005246509918, 6.70879647154145302141591820547, 7.63264412738571025222826742588, 8.881663628920254533601197892258, 9.183019396545750948990456086384

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