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Properties

Label 2-1840-5.4-c1-0-38
Degree $2$
Conductor $1840$
Sign $0.872 - 0.487i$
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  − 0.493i·3-s + (1.09 + 1.95i)5-s + 4.54i·7-s + 2.75·9-s + 4.61·11-s − 5.54i·13-s + (0.963 − 0.538i)15-s − 7.16i·17-s + 1.35·19-s + 2.24·21-s + i·23-s + (−2.62 + 4.25i)25-s − 2.84i·27-s + 3.66·29-s + 4.46·31-s + ⋯
L(s)  = 1  − 0.284i·3-s + (0.487 + 0.872i)5-s + 1.71i·7-s + 0.918·9-s + 1.39·11-s − 1.53i·13-s + (0.248 − 0.138i)15-s − 1.73i·17-s + 0.311·19-s + 0.489·21-s + 0.208i·23-s + (−0.524 + 0.851i)25-s − 0.546i·27-s + 0.680·29-s + 0.802·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.872 - 0.487i$
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: \((2,\ 1840,\ (\ :1/2),\ 0.872 - 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.334821829\)
\(L(\frac12)\) \(\approx\) \(2.334821829\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 - 1.95i)T \)
23 \( 1 - iT \)
good3 \( 1 + 0.493iT - 3T^{2} \)
7 \( 1 - 4.54iT - 7T^{2} \)
11 \( 1 - 4.61T + 11T^{2} \)
13 \( 1 + 5.54iT - 13T^{2} \)
17 \( 1 + 7.16iT - 17T^{2} \)
19 \( 1 - 1.35T + 19T^{2} \)
29 \( 1 - 3.66T + 29T^{2} \)
31 \( 1 - 4.46T + 31T^{2} \)
37 \( 1 + 3.32iT - 37T^{2} \)
41 \( 1 - 8.95T + 41T^{2} \)
43 \( 1 + 8.68iT - 43T^{2} \)
47 \( 1 - 9.59iT - 47T^{2} \)
53 \( 1 - 10.4iT - 53T^{2} \)
59 \( 1 + 2.08T + 59T^{2} \)
61 \( 1 + 0.686T + 61T^{2} \)
67 \( 1 - 8.84iT - 67T^{2} \)
71 \( 1 + 15.9T + 71T^{2} \)
73 \( 1 - 4.44iT - 73T^{2} \)
79 \( 1 + 8.65T + 79T^{2} \)
83 \( 1 - 4.43iT - 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 - 1.21iT - 97T^{2} \)
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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

−9.421676554095086342279939892169, −8.660678604794970653868525898774, −7.55543338010105455390422234410, −6.97912965757996088094440890595, −6.00778546024324982696267879585, −5.58119499990278834525605514812, −4.41662513932537918301333369956, −3.01177488729698419407858030344, −2.55489554039820855252291742561, −1.21701736352098300602706630844, 1.16418065116178685676121883825, 1.66403719009982070122144465228, 3.74503762558884843800383186956, 4.25564522822271914038059185543, 4.68911899635739988819396359004, 6.31331979718106477019792279207, 6.61592967654094444182946677587, 7.58499074491386105031945151093, 8.527274417571847033633213911534, 9.290990312845205695633436001692

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