Properties

Label 2-1840-1.1-c1-0-5
Degree $2$
Conductor $1840$
Sign $1$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.07·3-s + 5-s + 2.07·7-s + 6.48·9-s − 5.07·11-s − 3.48·13-s − 3.07·15-s + 6.48·17-s + 7.48·19-s − 6.40·21-s − 23-s + 25-s − 10.7·27-s − 1.56·29-s − 0.0791·31-s + 15.6·33-s + 2.07·35-s − 9.71·37-s + 10.7·39-s − 0.480·41-s − 8·43-s + 6.48·45-s − 6.96·47-s − 2.67·49-s − 19.9·51-s + 11.7·53-s − 5.07·55-s + ⋯
L(s)  = 1  − 1.77·3-s + 0.447·5-s + 0.785·7-s + 2.16·9-s − 1.53·11-s − 0.965·13-s − 0.795·15-s + 1.57·17-s + 1.71·19-s − 1.39·21-s − 0.208·23-s + 0.200·25-s − 2.06·27-s − 0.289·29-s − 0.0142·31-s + 2.72·33-s + 0.351·35-s − 1.59·37-s + 1.71·39-s − 0.0751·41-s − 1.21·43-s + 0.966·45-s − 1.01·47-s − 0.382·49-s − 2.79·51-s + 1.60·53-s − 0.684·55-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.9584000178\)
\(L(\frac12)\) \(\approx\) \(0.9584000178\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 + T \)
good3 \( 1 + 3.07T + 3T^{2} \)
7 \( 1 - 2.07T + 7T^{2} \)
11 \( 1 + 5.07T + 11T^{2} \)
13 \( 1 + 3.48T + 13T^{2} \)
17 \( 1 - 6.48T + 17T^{2} \)
19 \( 1 - 7.48T + 19T^{2} \)
29 \( 1 + 1.56T + 29T^{2} \)
31 \( 1 + 0.0791T + 31T^{2} \)
37 \( 1 + 9.71T + 37T^{2} \)
41 \( 1 + 0.480T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 6.96T + 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 - 7.88T + 61T^{2} \)
67 \( 1 - 9.71T + 67T^{2} \)
71 \( 1 - 9.67T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 - 4.59T + 83T^{2} \)
89 \( 1 - 8.31T + 89T^{2} \)
97 \( 1 - 7.23T + 97T^{2} \)
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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

−9.794656428668494269202341872315, −8.219588866434992146123950448801, −7.49365750999893591382265220268, −6.89249263015458572121416746154, −5.64539554487762014243368590645, −5.19421547260630377707680400286, −5.00024952232133145829992813807, −3.40060619205840993311204969511, −1.95445209653872842638962824825, −0.73536150366520661901555770619, 0.73536150366520661901555770619, 1.95445209653872842638962824825, 3.40060619205840993311204969511, 5.00024952232133145829992813807, 5.19421547260630377707680400286, 5.64539554487762014243368590645, 6.89249263015458572121416746154, 7.49365750999893591382265220268, 8.219588866434992146123950448801, 9.794656428668494269202341872315

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