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Properties

Label 2-1920-80.29-c1-0-47
Degree $2$
Conductor $1920$
Sign $-0.989 + 0.141i$
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 + (−0.607 − 2.15i)5-s + 2.25·7-s − 1.00i·9-s + (−1.66 + 1.66i)11-s + (−4.76 + 4.76i)13-s + (−1.95 − 1.09i)15-s − 6.99i·17-s + (−2.66 − 2.66i)19-s + (1.59 − 1.59i)21-s − 4.41·23-s + (−4.26 + 2.61i)25-s + (−0.707 − 0.707i)27-s + (−2.59 − 2.59i)29-s + 3.93·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.271 − 0.962i)5-s + 0.851·7-s − 0.333i·9-s + (−0.500 + 0.500i)11-s + (−1.32 + 1.32i)13-s + (−0.503 − 0.281i)15-s − 1.69i·17-s + (−0.611 − 0.611i)19-s + (0.347 − 0.347i)21-s − 0.921·23-s + (−0.852 + 0.522i)25-s + (−0.136 − 0.136i)27-s + (−0.481 − 0.481i)29-s + 0.706·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.989 + 0.141i$
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: \((2,\ 1920,\ (\ :1/2),\ -0.989 + 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8587465670\)
\(L(\frac12)\) \(\approx\) \(0.8587465670\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.607 + 2.15i)T \)
good7 \( 1 - 2.25T + 7T^{2} \)
11 \( 1 + (1.66 - 1.66i)T - 11iT^{2} \)
13 \( 1 + (4.76 - 4.76i)T - 13iT^{2} \)
17 \( 1 + 6.99iT - 17T^{2} \)
19 \( 1 + (2.66 + 2.66i)T + 19iT^{2} \)
23 \( 1 + 4.41T + 23T^{2} \)
29 \( 1 + (2.59 + 2.59i)T + 29iT^{2} \)
31 \( 1 - 3.93T + 31T^{2} \)
37 \( 1 + (2.01 + 2.01i)T + 37iT^{2} \)
41 \( 1 - 4.50iT - 41T^{2} \)
43 \( 1 + (7.14 + 7.14i)T + 43iT^{2} \)
47 \( 1 + 10.1iT - 47T^{2} \)
53 \( 1 + (-0.649 - 0.649i)T + 53iT^{2} \)
59 \( 1 + (-5.64 + 5.64i)T - 59iT^{2} \)
61 \( 1 + (-5.00 - 5.00i)T + 61iT^{2} \)
67 \( 1 + (4.95 - 4.95i)T - 67iT^{2} \)
71 \( 1 + 2.33iT - 71T^{2} \)
73 \( 1 - 2.18T + 73T^{2} \)
79 \( 1 + 6.38T + 79T^{2} \)
83 \( 1 + (-5.25 + 5.25i)T - 83iT^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 - 4.61iT - 97T^{2} \)
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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

−8.767150342886441339051273012836, −8.056075652423515191376074601219, −7.33395169676985864157389011213, −6.76239797337316970971956745904, −5.26758295623725181267275107936, −4.80152518643327751916523724680, −4.05357806239673566098578282776, −2.46721985403586154055923982072, −1.84030473455653979004650147382, −0.26829819274210948290147158906, 1.87681109050427704338251152078, 2.84745102432325070406782943554, 3.67566153963472723196225743187, 4.61581339200628411852755054102, 5.57557057183339770009234471093, 6.34015994319070081153259701147, 7.55291997376364718633657773315, 8.039047731910340819639519899658, 8.466526356022690290831467410584, 9.867899425124388836397523862081

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