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Properties

Label 2-1840-5.4-c1-0-40
Degree $2$
Conductor $1840$
Sign $0.447 + 0.894i$
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  − 2i·3-s + (2 − i)5-s + 3i·7-s − 9-s + 6i·13-s + (−2 − 4i)15-s − 7i·17-s − 4·19-s + 6·21-s − i·23-s + (3 − 4i)25-s − 4i·27-s + 9·29-s + 3·31-s + (3 + 6i)35-s + ⋯
L(s)  = 1  − 1.15i·3-s + (0.894 − 0.447i)5-s + 1.13i·7-s − 0.333·9-s + 1.66i·13-s + (−0.516 − 1.03i)15-s − 1.69i·17-s − 0.917·19-s + 1.30·21-s − 0.208i·23-s + (0.600 − 0.800i)25-s − 0.769i·27-s + 1.67·29-s + 0.538·31-s + (0.507 + 1.01i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.133489578\)
\(L(\frac12)\) \(\approx\) \(2.133489578\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
23 \( 1 + iT \)
good3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 7iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 7iT - 53T^{2} \)
59 \( 1 - 9T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 13iT - 67T^{2} \)
71 \( 1 - 13T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + 11iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.106880328259528099367033204138, −8.454109571540342954702242546988, −7.41921580403427661255451879341, −6.55897312714781837698762306884, −6.20508359996827198648627932013, −5.13324573187851431949735853493, −4.36357284262363655520652507468, −2.44472970356902299281897618415, −2.24498697136300666590928618800, −0.947255169095409542204540926973, 1.16587171903868198005473538032, 2.69073401660752880830005209203, 3.63720928975287075157645829245, 4.36214722805439604445009313855, 5.27769580530086317560682466396, 6.14661255766429408260920904217, 6.86338147016201561933453114650, 8.014136237444527536420658788535, 8.583204717326241163457665832685, 9.891853816327565069451934964429

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