Properties

Label 2-1840-460.459-c1-0-27
Degree $2$
Conductor $1840$
Sign $0.866 + 0.5i$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
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  − 2.23·5-s + 0.461i·7-s − 3·9-s − 0.799·17-s + 4.79i·23-s + 5.00·25-s + 9.78·29-s − 7.95i·31-s − 1.03i·35-s − 9.74·37-s + 5.78·41-s − 9.59i·43-s + 6.70·45-s + 6.78·49-s + 11.3·53-s + ⋯
L(s)  = 1  − 0.999·5-s + 0.174i·7-s − 9-s − 0.193·17-s + 0.999i·23-s + 1.00·25-s + 1.81·29-s − 1.42i·31-s − 0.174i·35-s − 1.60·37-s + 0.903·41-s − 1.46i·43-s + 0.999·45-s + 0.969·49-s + 1.55·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 0.866 + 0.5i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.077653043\)
\(L(\frac12)\) \(\approx\) \(1.077653043\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 2.23T \)
23 \( 1 - 4.79iT \)
good3 \( 1 + 3T^{2} \)
7 \( 1 - 0.461iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 0.799T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 - 9.78T + 29T^{2} \)
31 \( 1 + 7.95iT - 31T^{2} \)
37 \( 1 + 9.74T + 37T^{2} \)
41 \( 1 - 5.78T + 41T^{2} \)
43 \( 1 + 9.59iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 14.8iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 15.9iT - 67T^{2} \)
71 \( 1 - 5.89iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16.8iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 4.47T + 97T^{2} \)
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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

−8.854567773211461637543454934279, −8.548422769538562283833452224886, −7.63633152270606093696007327723, −6.93925747942996987170220713247, −5.91559134628600056162675666385, −5.14840071937098325417881719020, −4.12063232770484014830905726391, −3.30676245461652783243884568126, −2.31235608551701654622644966725, −0.57236667146099631994767377791, 0.833040937079540689133397217535, 2.58132555990661226949379539085, 3.36359262749177654624770690561, 4.40218631137485465959843743037, 5.11645630876112485702578405992, 6.24959921644727788722816014059, 6.95207699673284032033587983177, 7.83667331516870937102575390411, 8.613737275920439093474233918056, 8.951083333653540421855328871618

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