Properties

Label 2-1856-1.1-c3-0-17
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  + 2.34·3-s − 11.5·5-s + 8.54·7-s − 21.5·9-s − 47.4·11-s − 81.9·13-s − 27.0·15-s − 49.9·17-s + 20.4·19-s + 20.0·21-s + 105.·23-s + 8.57·25-s − 113.·27-s − 29·29-s + 22.4·31-s − 111.·33-s − 98.7·35-s − 316.·37-s − 192.·39-s − 427.·41-s + 190.·43-s + 248.·45-s + 459.·47-s − 270.·49-s − 117.·51-s + 545.·53-s + 548.·55-s + ⋯
L(s)  = 1  + 0.451·3-s − 1.03·5-s + 0.461·7-s − 0.796·9-s − 1.30·11-s − 1.74·13-s − 0.466·15-s − 0.712·17-s + 0.246·19-s + 0.208·21-s + 0.954·23-s + 0.0685·25-s − 0.810·27-s − 0.185·29-s + 0.130·31-s − 0.587·33-s − 0.476·35-s − 1.40·37-s − 0.788·39-s − 1.62·41-s + 0.676·43-s + 0.823·45-s + 1.42·47-s − 0.787·49-s − 0.321·51-s + 1.41·53-s + 1.34·55-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: $\chi_{1856} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6382091589\)
\(L(\frac12)\) \(\approx\) \(0.6382091589\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.34T + 27T^{2} \)
5 \( 1 + 11.5T + 125T^{2} \)
7 \( 1 - 8.54T + 343T^{2} \)
11 \( 1 + 47.4T + 1.33e3T^{2} \)
13 \( 1 + 81.9T + 2.19e3T^{2} \)
17 \( 1 + 49.9T + 4.91e3T^{2} \)
19 \( 1 - 20.4T + 6.85e3T^{2} \)
23 \( 1 - 105.T + 1.21e4T^{2} \)
31 \( 1 - 22.4T + 2.97e4T^{2} \)
37 \( 1 + 316.T + 5.06e4T^{2} \)
41 \( 1 + 427.T + 6.89e4T^{2} \)
43 \( 1 - 190.T + 7.95e4T^{2} \)
47 \( 1 - 459.T + 1.03e5T^{2} \)
53 \( 1 - 545.T + 1.48e5T^{2} \)
59 \( 1 + 809.T + 2.05e5T^{2} \)
61 \( 1 + 67.1T + 2.26e5T^{2} \)
67 \( 1 - 822.T + 3.00e5T^{2} \)
71 \( 1 - 666.T + 3.57e5T^{2} \)
73 \( 1 - 275.T + 3.89e5T^{2} \)
79 \( 1 - 681.T + 4.93e5T^{2} \)
83 \( 1 - 1.07e3T + 5.71e5T^{2} \)
89 \( 1 + 229.T + 7.04e5T^{2} \)
97 \( 1 - 261.T + 9.12e5T^{2} \)
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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

−8.744512100534240711612043331526, −7.977980251780114927282985793963, −7.56762070799236767131404192811, −6.77235779577515805391751300562, −5.25596221671263978639397733359, −4.99817105660888288217869270491, −3.81136600795429400933647184569, −2.84471090967733457552905287865, −2.16704098280529998691943726945, −0.33811193433676434060361259311, 0.33811193433676434060361259311, 2.16704098280529998691943726945, 2.84471090967733457552905287865, 3.81136600795429400933647184569, 4.99817105660888288217869270491, 5.25596221671263978639397733359, 6.77235779577515805391751300562, 7.56762070799236767131404192811, 7.977980251780114927282985793963, 8.744512100534240711612043331526

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