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Properties

Label 2-1840-5.4-c1-0-31
Degree $2$
Conductor $1840$
Sign $0.995 - 0.0938i$
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  + 0.580i·3-s + (−0.209 − 2.22i)5-s + 0.315i·7-s + 2.66·9-s + 4.34·11-s + 4.58i·13-s + (1.29 − 0.121i)15-s − 0.917i·17-s + 2.76·19-s − 0.182·21-s + i·23-s + (−4.91 + 0.933i)25-s + 3.28i·27-s − 7.03·29-s + 0.867·31-s + ⋯
L(s)  = 1  + 0.335i·3-s + (−0.0938 − 0.995i)5-s + 0.119i·7-s + 0.887·9-s + 1.31·11-s + 1.27i·13-s + (0.333 − 0.0314i)15-s − 0.222i·17-s + 0.634·19-s − 0.0399·21-s + 0.208i·23-s + (−0.982 + 0.186i)25-s + 0.632i·27-s − 1.30·29-s + 0.155·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.995 - 0.0938i$
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: \((2,\ 1840,\ (\ :1/2),\ 0.995 - 0.0938i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.019619012\)
\(L(\frac12)\) \(\approx\) \(2.019619012\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.209 + 2.22i)T \)
23 \( 1 - iT \)
good3 \( 1 - 0.580iT - 3T^{2} \)
7 \( 1 - 0.315iT - 7T^{2} \)
11 \( 1 - 4.34T + 11T^{2} \)
13 \( 1 - 4.58iT - 13T^{2} \)
17 \( 1 + 0.917iT - 17T^{2} \)
19 \( 1 - 2.76T + 19T^{2} \)
29 \( 1 + 7.03T + 29T^{2} \)
31 \( 1 - 0.867T + 31T^{2} \)
37 \( 1 + 4.68iT - 37T^{2} \)
41 \( 1 - 4.69T + 41T^{2} \)
43 \( 1 + 9.08iT - 43T^{2} \)
47 \( 1 - 8.24iT - 47T^{2} \)
53 \( 1 - 10.9iT - 53T^{2} \)
59 \( 1 - 1.50T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 1.68iT - 67T^{2} \)
71 \( 1 - 5.36T + 71T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 - 7.39T + 79T^{2} \)
83 \( 1 - 15.3iT - 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + 14.2iT - 97T^{2} \)
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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

−9.343660267973542050879925598470, −8.798039965566747127425499513874, −7.59364738678647366028615812988, −7.00094698525432743589670652647, −5.99164147330214635408141334956, −5.08544675293221395898432592117, −4.14421309768067775905920718015, −3.81991768666813546412031111495, −2.01478749951440108450856737493, −1.09074619162606516455815478407, 0.994225928135952564754297496753, 2.21913288680668041513862910580, 3.42548824719958414197157570818, 4.01435628543768837725678323114, 5.28988294614963044482685625726, 6.26765752684341133907792527717, 6.87897613103876338686662826336, 7.56021032597850412147044517930, 8.255977654527160604986001956067, 9.427488625035361465321521250331

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