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Properties

Label 2-1920-80.29-c1-0-27
Degree $2$
Conductor $1920$
Sign $-0.115 + 0.993i$
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 + (−1.95 + 1.07i)5-s − 1.22·7-s − 1.00i·9-s + (1.38 − 1.38i)11-s + (2.12 − 2.12i)13-s + (−0.623 + 2.14i)15-s + 6.00i·17-s + (−3.06 − 3.06i)19-s + (−0.863 + 0.863i)21-s + 2.90·23-s + (2.67 − 4.22i)25-s + (−0.707 − 0.707i)27-s + (−3.18 − 3.18i)29-s + 3.88·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.876 + 0.481i)5-s − 0.461·7-s − 0.333i·9-s + (0.416 − 0.416i)11-s + (0.588 − 0.588i)13-s + (−0.161 + 0.554i)15-s + 1.45i·17-s + (−0.702 − 0.702i)19-s + (−0.188 + 0.188i)21-s + 0.606·23-s + (0.535 − 0.844i)25-s + (−0.136 − 0.136i)27-s + (−0.591 − 0.591i)29-s + 0.697·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.115 + 0.993i$
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: \((2,\ 1920,\ (\ :1/2),\ -0.115 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.219179642\)
\(L(\frac12)\) \(\approx\) \(1.219179642\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.95 - 1.07i)T \)
good7 \( 1 + 1.22T + 7T^{2} \)
11 \( 1 + (-1.38 + 1.38i)T - 11iT^{2} \)
13 \( 1 + (-2.12 + 2.12i)T - 13iT^{2} \)
17 \( 1 - 6.00iT - 17T^{2} \)
19 \( 1 + (3.06 + 3.06i)T + 19iT^{2} \)
23 \( 1 - 2.90T + 23T^{2} \)
29 \( 1 + (3.18 + 3.18i)T + 29iT^{2} \)
31 \( 1 - 3.88T + 31T^{2} \)
37 \( 1 + (-2.44 - 2.44i)T + 37iT^{2} \)
41 \( 1 + 2.38iT - 41T^{2} \)
43 \( 1 + (9.00 + 9.00i)T + 43iT^{2} \)
47 \( 1 + 0.586iT - 47T^{2} \)
53 \( 1 + (2.36 + 2.36i)T + 53iT^{2} \)
59 \( 1 + (-8.43 + 8.43i)T - 59iT^{2} \)
61 \( 1 + (9.98 + 9.98i)T + 61iT^{2} \)
67 \( 1 + (-3.82 + 3.82i)T - 67iT^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + 1.31T + 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 + (-2.91 + 2.91i)T - 83iT^{2} \)
89 \( 1 + 9.58iT - 89T^{2} \)
97 \( 1 + 9.45iT - 97T^{2} \)
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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

−8.665737430858030808999805493915, −8.315429637214921085493044480117, −7.50113068694434643095908484960, −6.50503475410832318793359950529, −6.21745549835148667835071243816, −4.79508493526354773090155802355, −3.64550603778635716625085128042, −3.29977738217084306323458158056, −1.98656037071238276382112275826, −0.45683596645905253181059454093, 1.26206385702424235391675643911, 2.75126705582637708473225797715, 3.67822432310002275385920466302, 4.40182577470296921271409546664, 5.14008376547862054275088007847, 6.37594480699218459320957407375, 7.12833694981506701213914892945, 7.940219333382801339242215833416, 8.737670356376351475225962603638, 9.311041082057662976715902905094

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