Properties

Label 2-1856-1.1-c3-0-126
Degree $2$
Conductor $1856$
Sign $1$
Analytic cond. $109.507$
Root an. cond. $10.4645$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.17·3-s + 14.0·5-s + 13.6·7-s + 57.2·9-s + 4.50·11-s − 7.29·13-s + 128.·15-s + 28.0·17-s + 100.·19-s + 125.·21-s − 62.4·23-s + 71.2·25-s + 277.·27-s + 29·29-s − 156.·31-s + 41.3·33-s + 191.·35-s + 233.·37-s − 66.9·39-s + 7.39·41-s + 31.0·43-s + 801.·45-s − 226.·47-s − 156.·49-s + 257.·51-s − 254.·53-s + 63.1·55-s + ⋯
L(s)  = 1  + 1.76·3-s + 1.25·5-s + 0.737·7-s + 2.11·9-s + 0.123·11-s − 0.155·13-s + 2.21·15-s + 0.399·17-s + 1.20·19-s + 1.30·21-s − 0.566·23-s + 0.570·25-s + 1.97·27-s + 0.185·29-s − 0.909·31-s + 0.218·33-s + 0.924·35-s + 1.03·37-s − 0.274·39-s + 0.0281·41-s + 0.110·43-s + 2.65·45-s − 0.703·47-s − 0.456·49-s + 0.705·51-s − 0.658·53-s + 0.154·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $1$
Analytic conductor: \(109.507\)
Root analytic conductor: \(10.4645\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.880561806\)
\(L(\frac12)\) \(\approx\) \(6.880561806\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 - 29T \)
good3 \( 1 - 9.17T + 27T^{2} \)
5 \( 1 - 14.0T + 125T^{2} \)
7 \( 1 - 13.6T + 343T^{2} \)
11 \( 1 - 4.50T + 1.33e3T^{2} \)
13 \( 1 + 7.29T + 2.19e3T^{2} \)
17 \( 1 - 28.0T + 4.91e3T^{2} \)
19 \( 1 - 100.T + 6.85e3T^{2} \)
23 \( 1 + 62.4T + 1.21e4T^{2} \)
31 \( 1 + 156.T + 2.97e4T^{2} \)
37 \( 1 - 233.T + 5.06e4T^{2} \)
41 \( 1 - 7.39T + 6.89e4T^{2} \)
43 \( 1 - 31.0T + 7.95e4T^{2} \)
47 \( 1 + 226.T + 1.03e5T^{2} \)
53 \( 1 + 254.T + 1.48e5T^{2} \)
59 \( 1 + 33.2T + 2.05e5T^{2} \)
61 \( 1 + 644.T + 2.26e5T^{2} \)
67 \( 1 - 939.T + 3.00e5T^{2} \)
71 \( 1 + 842.T + 3.57e5T^{2} \)
73 \( 1 - 664.T + 3.89e5T^{2} \)
79 \( 1 - 0.292T + 4.93e5T^{2} \)
83 \( 1 - 283.T + 5.71e5T^{2} \)
89 \( 1 - 1.41e3T + 7.04e5T^{2} \)
97 \( 1 + 642.T + 9.12e5T^{2} \)
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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

−9.029353822239741803094619135885, −8.006300905147196101340448091883, −7.69480295491587660102890373979, −6.63701958941021388454463887201, −5.60474408670308329768360310625, −4.72594245200637812771188703787, −3.65617295395862375634501029218, −2.79236278879885644449442708859, −1.95449338567705054928021922037, −1.30053867833231042195990594753, 1.30053867833231042195990594753, 1.95449338567705054928021922037, 2.79236278879885644449442708859, 3.65617295395862375634501029218, 4.72594245200637812771188703787, 5.60474408670308329768360310625, 6.63701958941021388454463887201, 7.69480295491587660102890373979, 8.006300905147196101340448091883, 9.029353822239741803094619135885

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