Properties

Label 2-1840-1.1-c1-0-18
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.66·3-s − 5-s − 3.66·7-s + 4.12·9-s + 1.21·11-s + 2.21·13-s + 2.66·15-s + 1.21·17-s + 2.57·19-s + 9.79·21-s − 23-s + 25-s − 3.00·27-s + 1.45·29-s + 6.46·31-s − 3.24·33-s + 3.66·35-s + 4·37-s − 5.90·39-s − 10.9·41-s − 6.90·43-s − 4.12·45-s − 5.45·47-s + 6.46·49-s − 3.24·51-s + 3.81·53-s − 1.21·55-s + ⋯
L(s)  = 1  − 1.54·3-s − 0.447·5-s − 1.38·7-s + 1.37·9-s + 0.366·11-s + 0.614·13-s + 0.689·15-s + 0.294·17-s + 0.591·19-s + 2.13·21-s − 0.208·23-s + 0.200·25-s − 0.577·27-s + 0.270·29-s + 1.16·31-s − 0.564·33-s + 0.620·35-s + 0.657·37-s − 0.946·39-s − 1.70·41-s − 1.05·43-s − 0.614·45-s − 0.795·47-s + 0.923·49-s − 0.453·51-s + 0.524·53-s − 0.163·55-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: \(1\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.66T + 3T^{2} \)
7 \( 1 + 3.66T + 7T^{2} \)
11 \( 1 - 1.21T + 11T^{2} \)
13 \( 1 - 2.21T + 13T^{2} \)
17 \( 1 - 1.21T + 17T^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
29 \( 1 - 1.45T + 29T^{2} \)
31 \( 1 - 6.46T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 6.90T + 43T^{2} \)
47 \( 1 + 5.45T + 47T^{2} \)
53 \( 1 - 3.81T + 53T^{2} \)
59 \( 1 - 4.24T + 59T^{2} \)
61 \( 1 - 6.78T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 - 6.91T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 - 1.69T + 97T^{2} \)
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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

−8.954024069073385062795153787427, −7.988367821801163022835933366548, −6.82377209018436714601766899280, −6.52497699449911203536753541489, −5.74828110519772520210281619439, −4.90408229360456541022233477896, −3.89303622839608625433553678016, −3.03818114620558777499039094258, −1.16317780908747699667837173688, 0, 1.16317780908747699667837173688, 3.03818114620558777499039094258, 3.89303622839608625433553678016, 4.90408229360456541022233477896, 5.74828110519772520210281619439, 6.52497699449911203536753541489, 6.82377209018436714601766899280, 7.988367821801163022835933366548, 8.954024069073385062795153787427

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