Properties

Label 2-1840-460.459-c1-0-25
Degree $2$
Conductor $1840$
Sign $0.995 - 0.0960i$
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  − 3-s + (1.22 − 1.87i)5-s − 2·9-s + 4.89·11-s + 4.58i·13-s + (−1.22 + 1.87i)15-s − 2.44·17-s + 2.44·19-s + (3 + 3.74i)23-s + (−2 − 4.58i)25-s + 5·27-s − 3·29-s + 4.58i·31-s − 4.89·33-s + 4.89·37-s + ⋯
L(s)  = 1  − 0.577·3-s + (0.547 − 0.836i)5-s − 0.666·9-s + 1.47·11-s + 1.27i·13-s + (−0.316 + 0.483i)15-s − 0.594·17-s + 0.561·19-s + (0.625 + 0.780i)23-s + (−0.400 − 0.916i)25-s + 0.962·27-s − 0.557·29-s + 0.823i·31-s − 0.852·33-s + 0.805·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.995 - 0.0960i$
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: \((2,\ 1840,\ (\ :1/2),\ 0.995 - 0.0960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.561692599\)
\(L(\frac12)\) \(\approx\) \(1.561692599\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 + 1.87i)T \)
23 \( 1 + (-3 - 3.74i)T \)
good3 \( 1 + T + 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 - 4.58iT - 13T^{2} \)
17 \( 1 + 2.44T + 17T^{2} \)
19 \( 1 - 2.44T + 19T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 4.58iT - 31T^{2} \)
37 \( 1 - 4.89T + 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 11.2iT - 43T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 + 4.89T + 53T^{2} \)
59 \( 1 + 9.16iT - 59T^{2} \)
61 \( 1 + 11.2iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 13.7iT - 71T^{2} \)
73 \( 1 - 4.58iT - 73T^{2} \)
79 \( 1 - 7.34T + 79T^{2} \)
83 \( 1 + 3.74iT - 83T^{2} \)
89 \( 1 - 3.74iT - 89T^{2} \)
97 \( 1 - 9.79T + 97T^{2} \)
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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

−9.091177449394621344324925454014, −8.899188007601785130611387029035, −7.66932724049607304266922637517, −6.54595771715411699149475295052, −6.20739285803104364496575028317, −5.16056666922376986778161543478, −4.52120335406798510341075795911, −3.47630767767347573219527445396, −2.00174217957129748390936091453, −1.01524059167307022167846659147, 0.803299423139157110648410327211, 2.33621672628916130765958438838, 3.23218360042330779864872587638, 4.25885469736385794446059668800, 5.55426292876499768277845738675, 5.90857152731437429865548924810, 6.79352550110373778663312746743, 7.43041231224751264075505244391, 8.639870659064245570933409725654, 9.196085799211912597387110951917

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