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Properties

Label 2-1920-80.29-c1-0-34
Degree $2$
Conductor $1920$
Sign $0.256 + 0.966i$
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  + (0.707 − 0.707i)3-s + (1.86 − 1.24i)5-s − 1.58·7-s − 1.00i·9-s + (3.92 − 3.92i)11-s + (−3.10 + 3.10i)13-s + (0.438 − 2.19i)15-s − 1.48i·17-s + (4.94 + 4.94i)19-s + (−1.12 + 1.12i)21-s + 6.61·23-s + (1.92 − 4.61i)25-s + (−0.707 − 0.707i)27-s + (−4.42 − 4.42i)29-s + 1.50·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.831 − 0.554i)5-s − 0.600·7-s − 0.333i·9-s + (1.18 − 1.18i)11-s + (−0.861 + 0.861i)13-s + (0.113 − 0.566i)15-s − 0.359i·17-s + (1.13 + 1.13i)19-s + (−0.245 + 0.245i)21-s + 1.38·23-s + (0.384 − 0.923i)25-s + (−0.136 − 0.136i)27-s + (−0.821 − 0.821i)29-s + 0.270·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.256 + 0.966i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.256 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.308966992\)
\(L(\frac12)\) \(\approx\) \(2.308966992\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-1.86 + 1.24i)T \)
good7 \( 1 + 1.58T + 7T^{2} \)
11 \( 1 + (-3.92 + 3.92i)T - 11iT^{2} \)
13 \( 1 + (3.10 - 3.10i)T - 13iT^{2} \)
17 \( 1 + 1.48iT - 17T^{2} \)
19 \( 1 + (-4.94 - 4.94i)T + 19iT^{2} \)
23 \( 1 - 6.61T + 23T^{2} \)
29 \( 1 + (4.42 + 4.42i)T + 29iT^{2} \)
31 \( 1 - 1.50T + 31T^{2} \)
37 \( 1 + (-2.14 - 2.14i)T + 37iT^{2} \)
41 \( 1 + 6.84iT - 41T^{2} \)
43 \( 1 + (-0.322 - 0.322i)T + 43iT^{2} \)
47 \( 1 + 13.3iT - 47T^{2} \)
53 \( 1 + (-0.931 - 0.931i)T + 53iT^{2} \)
59 \( 1 + (-1.14 + 1.14i)T - 59iT^{2} \)
61 \( 1 + (2.67 + 2.67i)T + 61iT^{2} \)
67 \( 1 + (5.43 - 5.43i)T - 67iT^{2} \)
71 \( 1 + 2.26iT - 71T^{2} \)
73 \( 1 + 5.27T + 73T^{2} \)
79 \( 1 - 6.52T + 79T^{2} \)
83 \( 1 + (0.973 - 0.973i)T - 83iT^{2} \)
89 \( 1 - 6.83iT - 89T^{2} \)
97 \( 1 + 15.8iT - 97T^{2} \)
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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

−9.189061751169891298948969927924, −8.447964320125013480799171389966, −7.38347678468580218112886814715, −6.64598271680335689674484083350, −5.93553912524477238661851739940, −5.12665578039932044729194880543, −3.91720340176137105693903258596, −3.06530335372007281594136911837, −1.91353545681186390392705070629, −0.863362008700964738225890536668, 1.41788759676870446875130292665, 2.72228494998223404515327677674, 3.23712744789466999352516034790, 4.53218149141116142336950630798, 5.24838420975254457241675725771, 6.28779164734462729240353671143, 7.08177333086841899285281484264, 7.55569599983288648826967832199, 8.974408218698349004434261934944, 9.507728375553146134451812243639

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