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Properties

Label 2-1840-460.459-c1-0-37
Degree $2$
Conductor $1840$
Sign $0.838 - 0.544i$
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  + 2.49·3-s + (−2.19 + 0.417i)5-s + 3.04i·7-s + 3.24·9-s + 5.87·11-s − 6.00i·13-s + (−5.48 + 1.04i)15-s − 0.968·17-s + 1.72·19-s + 7.61i·21-s + (3.79 + 2.93i)23-s + (4.65 − 1.83i)25-s + 0.611·27-s − 3.89·29-s + 9.01i·31-s + ⋯
L(s)  = 1  + 1.44·3-s + (−0.982 + 0.186i)5-s + 1.15i·7-s + 1.08·9-s + 1.77·11-s − 1.66i·13-s + (−1.41 + 0.269i)15-s − 0.234·17-s + 0.395·19-s + 1.66i·21-s + (0.790 + 0.612i)23-s + (0.930 − 0.366i)25-s + 0.117·27-s − 0.723·29-s + 1.61i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.838 - 0.544i$
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: \((2,\ 1840,\ (\ :1/2),\ 0.838 - 0.544i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.706911911\)
\(L(\frac12)\) \(\approx\) \(2.706911911\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 - 0.417i)T \)
23 \( 1 + (-3.79 - 2.93i)T \)
good3 \( 1 - 2.49T + 3T^{2} \)
7 \( 1 - 3.04iT - 7T^{2} \)
11 \( 1 - 5.87T + 11T^{2} \)
13 \( 1 + 6.00iT - 13T^{2} \)
17 \( 1 + 0.968T + 17T^{2} \)
19 \( 1 - 1.72T + 19T^{2} \)
29 \( 1 + 3.89T + 29T^{2} \)
31 \( 1 - 9.01iT - 31T^{2} \)
37 \( 1 - 2.15T + 37T^{2} \)
41 \( 1 - 6.17T + 41T^{2} \)
43 \( 1 - 9.08iT - 43T^{2} \)
47 \( 1 - 9.73T + 47T^{2} \)
53 \( 1 + 9.18T + 53T^{2} \)
59 \( 1 + 1.17iT - 59T^{2} \)
61 \( 1 + 0.951iT - 61T^{2} \)
67 \( 1 + 9.83iT - 67T^{2} \)
71 \( 1 - 8.99iT - 71T^{2} \)
73 \( 1 - 8.53iT - 73T^{2} \)
79 \( 1 - 5.59T + 79T^{2} \)
83 \( 1 + 5.97iT - 83T^{2} \)
89 \( 1 - 7.22iT - 89T^{2} \)
97 \( 1 + 7.51T + 97T^{2} \)
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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

−9.072772151657443935046135204777, −8.643443204871224416065563695489, −7.87414734223587317803264898222, −7.26270130311485672437116119467, −6.23294616533643152190010685651, −5.18941427591627153270941301648, −4.01961330746296770178736921267, −3.25661424096906505587966932286, −2.73557107407778948600319350220, −1.29414946137425113246976746675, 1.01303923341071817799548880710, 2.19135552046920538663690214271, 3.56395998623437067887478159780, 4.03323094673539112953328620970, 4.47962838254401497587695703059, 6.31425404152978934455571030917, 7.21265780556528188258533258981, 7.46468420892138495569198602202, 8.520859263515472766650647515016, 9.247667608310359693978289252656

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